Irving  Stringham 


L1PPINCOTTS 
ELEMENTARY  ARITHMETIC 

EMBRACING 

THE   SCIENCE   AND   PRACTICAL   APPLICATIONS 
OF   NUMBERS 


BY 

J.  MORGAN   RAWLINS,  A.M. 

// 

AUTHOR  OF  "  LIPPINCOTT'S  PRACTICAL  ARITHMETIC"  AND  "  LIPPINCOTT'S 
MENTAL  ARITHMETIC" 


PHILADELPHIA 

J.  B.  LIPPINCOTT    COMPANY 


BY 

J,  B.  LIPPINCOTT  COMPANY. 


ElECTROTYPED  AND    PRINTED   BY  J.  B.  LiPPINCOTT   COMPANY,  PHILADELPHIA,  U.  S.  A. 


PREFACE. 


"  IN  all  the  affairs  of  life,  the  arithmetical  part  of  the 
business  is  the  dominant  one."  How  many  and  how 
much  have  we  ?  How  many  and  how  much  do  we  want? 
are  questions  that  constantly  ohtrude  themselves  for  an- 
swer. "  Arithmetic  is  the  conclusive  science  that  men 
have  to  apply,  all  their  days,  to  all  their  affairs."  Of  all 
the  sciences,  therefore,  with  which  men  have  to  do,  none 
deserves  more  intelligent  study  or  more  prudent  appli- 
cation than  the  arithmetical  one.  Memory  alone  cannot 
deal  with  it  or  comprehend  it.  Pupils  must  he  trained  to 
see,  to  hear,  to  think,  in  order  to  grasp  its  truths  and  learn 
to  apply  them.  We  dare  say  that  the  philosophical  study 
of  no  other  subject  will  impart  to  the  mind  of  youth  a 
higher  degree  of  acuteness  and  penetration.  u  It  makes 
men  subtile,"  said  Lord  Bacon. 

Very  few  pupils,  we  presume,  enter  school  for  the  first 
time  who  have  not  some  idea  of  number  and  of  nu- 
merical combinations;  but  this  knowledge,  incidentally 
gathered  here  and  there,  necessarily  lies  in  the  mind  in  no 
definite  or  connected  order.  It  is  important,  therefore, 
that  when  pupils  begin  the  systematic  study  of  Arith- 
metic, they  begin  with  the  very  first  lesson,  so  that  what 
they  already  know  may  be  set  in  order  and  be  made  the 
basis  of  what  is  next  to  be  learned.  Beginners  should  be 

800552 


IV  PREFACE 

led  without  delay  to  perceive  that  the  lesson  learned  to- 
day is  little  more  thaii  the  cultivated  ground  out  of  which 
is  to  grow  the  lesson  of  to-morrow. 

It  is  a  fatal  error,  only  too  common,  to  start  a  child  to 
study  where  that  which  he  is  asked  to  learn  is  out  of 
touch  with  that  which  he  already  knows.  Pupils  should 
be  taught  very  early  to  keep  an  accurate  separation  of 
the  known  from  the  unknown,  and  "to  he  careful  not  to 
stamp  a  thing  as  known"  until  they  have  fully  mastered 
it  in  all  its  relations  to  that  which  they  know,  and  have 
done  so  " in  that  way  which  conscience  calls  honest" 

The  following  Elementary  Treatise  on  Arithmetic  has 
been  prepared  with  the  view  of  presenting  to  both  teacher 
and  pupil  a  thoroughly  systematic  and  gently  graded 
scheme  in  which  they  may  together  make  daily  progress 
in  scientific  knowledge  of  the  subject,  and  by  a  mutual 
interest  in  the  work  gather  by  diligence  many  of  the  best 
fruits  of  industry. 

Nothing,  from  beginning  to  end,  has  been  written  as 
mere  verbiage,  undeserving  of  attention.  Every  word 
has  a  measure  of  significance,  and  every  sentence  in  the 
book  is  there  for  the  single  purpose  of  being  understood. 

J.  M.  R. 


CONTENTS. 


Idea  of  Number  and  Primary  Processes 


PAGE 
1-40 


PART    II. 


Definitions 41 

Notation   and    Numera- 
tion      42 

Periods  and  Orders   ....  49 

Decimal  Notation      ....  58 

United  States  Money  ...  60 

Roman  Notation 62 

Addition 65 

Single  Columns 69 

Several  Columns 74 

Subtraction 80 

Figures  of  Minuend  all  of 

Greater  Value    ....  84 
Figures  of  Less  Value  in  the 

Minuend 87 

Addition   and   Subtrac- 
tion      92 

Review 94 

Multiplication 98 

Multiplier  a  Single  Figure  .  103 
Multiplier  with  Ciphers  An- 
nexed .  105 


Multiplication !                    PAGE 
Multiplier  Two  or  More  Sig- 
nificant Figures     .    .    .  107 

Oral  Review 110 

Written  Review Ill 

Division 114 

Equal  Parts 115 

Divisor  a  Single  Digit .    .    .  119 
Divisor  a  Single  Digit  with 

Ciphers  Annexed  .    .    .  123 
Divisor    Any    Number    of 

Digits 125 

Parenthesis  and  Vinculum  .  131 

Analysis 132 

Review 136 

Factoring 137 

Cancellation 140 

Fractions 144 

Reduction 148 

To  Lowest  Terms     ...  149 

To  Higher  Terms  ....  150 
To  Common  Denominator    151 

To  Mixed  Numbers  .        ,  153 


VI 


CONTENTS 


Fractions :  PAGE 
Reduction  of  Mixed  Num- 
bers    154 

Addition  of  Fractions  .    .    .  156 

Subtraction  of  Fractions     .  161 

Multiplication  of  Fractions  165 

Division  of  Fractions  ...  172 

Complex  Fractions  ....  180 

Fractional  Relation  ....  181 

Oral  Review 184 

Written  Review 185 

Decimal  Fractions    ...  189 

Reduction  of  Decimals    .    .  194 

Addition  of  Decimals  .    .    .  197 

Subtraction  of  Decimals  .    .  199 

Multiplication  of  Decimals .  201 

Division  of  Decimals    .    .    .  204 

United  States  Money  ...  210 

Bills  and  Accounts.   .   .  216 

Denominate  Numbers    .  220 

Linear  Measure     ....  221 

Surface  Measure    ....  222 

Dry  Measure 225 


Denominate  Numbers:  PAGE 

Avoirdupois  Weight    .    .  226 

Troy  Weight 227 

Apothecaries'  Weight  .    .  228 

Divisions  of  Time     ...  229 

Counting 230 

Stationers'  Table  ....  231 
Reduction    of    Denominate 

Numbers 232 

Addition     of     Denominate 

Numbers 238 

Subtraction  of  Denominate 

Numbers 239 

Multiplication   of  Denomi- 
nate Numbers    ....  240 
Division     of     Denominate 

Numbers 241 

Measurement  of  Surfaces    .  242 

Measurement  of  Solids    .    .  245 

Board  Measure      247 

Percentage 249 

Interest 252 

General  Review .....  256 


GENERAL  SUGGESTIONS. 


PART  I.  of  this  book  presupposes  the  occasional  neces- 
sity of  introducing  pupils  to  the  first  rudiments  of  the 
subject. 

In  developing  the  idea  of  number,  and  teaching  the 
simple  numerical  operations,  real  objects  are  greatly  to  be 
preferred  to  pictures,  however  artistic  and  striking  they 
may  be.  But  what  is  to  be  preferred  to  all  other  agencies, 
whether  pictures  or  real  objects,  or  even  the  book  itself, 
is  the  voice  and  action  of  the  live  and  intelligent  teacher, 
without  which  little  of  educational  value  is  ever  accom- 
plished in  school.  Object  lessons,  however,  are  of  great 
value  in  illustrating  and  impressing  the  teacher's  mean- 
ing; but  no  objects  should  be  used  except  those  of  simple 
form  and  construction,  lest  the  mind  of  the  pupil  be  di- 
verted by  them  from  the  primary  object  in  view,  namely, 
the  inculcation  of  the  idea  of  number  and  of  numerical 
combinations. 

When  objects  are  employed,  the  youthful  pupil  should 
be  allowed  to  take  them  in  his  own  hands  and  give  proof 
of  knowledge  gained  by  showing  without  help  how  and 
what  they  explain.  This  will  be  to  his  liking,  and  liking 
is  a  supreme  element  of  learning. 

The  successful  teacher  is  not  he  who  does  both  his  own 
and  the  pupil's  work,  but  he  who  best  directs  the  pupil's 

vii 


viii  GENERAL  SUGGESTIONS 

activities,  leads  him  to  love  learning,  and  to  overcome 
difficulties  by  his  own  efforts. 

In  reciting,  pupils  should  be  required  to  give  their 
answers  in  complete  sentences,  and  that,  too,  without 
hesitation  or  counting.  They  should  be  able  to  say  "  4 
arid  3  are  7,"  or  "  4  less  3  equals  1,"  as  promptly  and  with 
as  little  apparent  effort  as  they  would  spell  a  word  of 
three  letters.  In  sight  exercises,  where  rapidity  is  the 
object  sought,  results  alone  should  be  given ;  thus : 
"seven,"  "one." 

The  use  of  concert  exercises  should  not  be  constant, 
but  only  occasional,  as  a  ready  means  to  give  diversity  or 
to  revive  flagging  interest.  As  far  as  possible,  the  mem- 
bers of  a  class  should  be  drilled  individually,  for  each  has 
a  separate  and  distinct  individuality  that  demands  and 
must  receive  from  the  teacher  the  carefulest  recognition. 
Every  exercise  should  be  made  so  interesting  as  to  engage 
the  undivided  attention  of  every  pupil,  and  to  effect  this 
they  must  of  necessity  be  lively,  varied,  and  above  all 
brief.  The  very  best  judgment  of  the  teacher  is  de- 
manded here.  The  processes  of  Addition  and  Subtraction 
are  so  intimately  related  that  they  should  be  taught  at  first 
together.  In  like  manner  should  be  taught  the  closely 
related  operations  of  Multiplication  and  Division. 

All  written  work  should  be  as  neat  and  well  arranged 
as  the  pupils  are  capable  of  doing  it.  Ill-formed  figures 
and  careless  arrangement  are  fruitful  sources  of  error  in 
results. 

Pupils  while  in  school  should  be  kept  constantly  em- 
ployed. They  must  do  if  they  would  learn.  Idleness 
should  not  be  tolerated  for  a  moment.  The  forward 


GENERAL  SUGGESTIONS  ix 

movement  should  be  frequently  halted,  and  past  lessons 
repeated.  It  is  by  repetition  that  acquisition  becomes 
fixed  and  sure :  the  pupil's  ignorance  never  ceases  to  be 
the  teacher's  opportunity. 

Pupils  should  be  drilled  on  all  exercises  with  great  care 
and  thoroughness.  The  primary  lessons  of  the  book  are 
presented  as  suggestive  rather  than  as  exhaustive,  and  the 
teacher  will  often  find  it  necessary  to  supplement  them. 
In  such  cases  the  suggestions  at  the  foot  of  the  pages  will 
be  helpful. 


ELEMENTARY  ARITHMETIC 


PART    I. 

IDEA  OF  NUMBER  AND  PRIMARY  PROCESSES- 


LESSON  i. 


1.  How  many  sheep  on  the  left  ? 

2.  How  many  on  the  right  ? 

3.  How  many  sheep  are  one  sheep  and  one  sheep  ? 

4.  Hold  up  one  finger.    Hold  up  another  finger.    How 
many  fingers  are  you  now  holding  up  ? 

5.  How  many  are  one  and  one  ? 

6.  One  sheep  and  two  sheep  are  how  many  sheep? 
How  many  are  two  and  one  ? 

7.  Instead  of  writing  the  words 
one,  two,  and  three,  we  may  use  fig- 
ures.    Now,  make  them  neatly. 

NOTE.— Drill  upon  1,  2,  and  3  as  numbers,  not  as  figures.  Teach  Ad- 
dition and  Subtraction  at  the  same  time :  2  and  1  are  3 ;  therefore,  1  from  3 
leaves  2,  and  2  from  3  leaves  1. 

1 


One. 

9           S 
Two.     Three. 

•-  3KLEMENTARY   ARITHMETIC 


LESSON    II. 
Pour  and  Combinations. 

1.  How  many  hands  have  you  ? 
How  many  feet  ? 

2.  Hold  up  as  many  fingers  as 
there  are  stars  on  the  left. 

3.  How  many  stars  on  the  left?     How  many  on  the 
right  ?     Three  stars  and  one  star  are  how  many  stars  ? 

4.  Take  four  objects  from  the  table.     Return  one  to 
the  table.     How  many  have  you  now  ?     One  taken  from 
four  leaves  how  many  ? 

5.  Take  four  objects.    Give  two  to  your  teacher.    How 
many  have  you  now  ?     Two  taken  from  four  leaves  how 
many? 

6.  Make  four  short  lines  on  the  blackboard.     Place 
your  hand  over  three  of  the  lines.     How  many  do  you 
see  ?     Three  from  four  leaves  how  many  ? 

7.  One  star,  and  one  star,  and  one  star,  are  how  many 
stars  ?     How  many  ones  in  three  ?     How  many  are  three 
ones? 

8.  How  many  boys  and  girls  in  the  class  can  take  ob- 
jects and  show  that 

1  and  1  are  2.  1  taken  from  2  leaves  1. 

1  and  2  are  3.  1  taken  from  3  leaves  2. 

1  and  3  are  4.  1  taken  from  4  leaves  3. 

2  and  2  are  4.  2  taken  from  3  leaves  1. 

To  THE  TEACHER. — Exercises  similar  to 
the  foregoing  should  be  given  until  pupils 
can  read  the  combinations— that  is,  state  their 
value— as  promptly  as  they  say  "at"  when 
they  see  the  letters  a-t. 


ELEMENTARY  ARITHMETIC 


LESSON    III. 
Five  and  Combinations. 

1.  Look  at  the  stars  on 
the  left  and  tell  how  many  ? 

2.  How   many  twos    in 
four? 

3.  Look  at  the  stars  on  the  right  and  tell  how  many 
more  stars  are  there  than  on  the  left. 

4.  How  many  are  four  and  one  ?     Three  and  two  ? 

5.  How  many  circles  are  Q  O  O  an(*  O  •    How  many 
are  Q  O  and  O  O  ? 

6.  Here  are  some  groups  of  stars.     At  sight  name  the 
number  in  each. 

#1      ft  #  2      ft  ft  ft  3      ###3      ft  ft  ft  &  4      ft  1 
A  ft  _2_     ftft_2^         *     _!_       ftft   J2_  *       JL     ^JL 

¥4  4  5  52 

7.  Examine  the  groups  and  tell  how  each  is  made  up. 
Name  the  number  in  each  row.     Name  the  number  in 
each  group. 

8.  If  you  were  to  use  figures  for  the  stars,  you  would 
have :  1         2         3         3,4         1 

221211 

9.  We  have  learned  a  new  number  to-day,  and  must 
know  how  to  say  five  with  a  figure.    Here  is  the  figure  5. 
Try  to  make  it  on  your  slate. 

To  THE  TEACHER. — Arrange  groups  of 
objects  from  one  to  five  and  practice  by  hav- 
ing pupils  name  the  number  in  each  group. 
Pupils  should  recognize  at  sight  the  number 
of  objects  in  each  group. 


ELEMENTARY  ARITHMETIC 


LESSON    IV. 
Six  and  Combinations. 

1.  How  many  are  three 
and  two  ? 

2.  How   many  are  five 
and  one  ? 

3.  Take  three  objects  from  the  table.     Take  three  in 
your  other  hand.     How  many  have  you  in  both  hands  ? 
How  many  are  three  and  three?     How  many  are  two 
and  three  ? 

4.  Pass  one  of  those  in  your  right  hand  to  your  left 
hand.     How  many  have  you  in  each  hand  now?     How 
many  in  both  hands?     Four  and  two  are  how  many? 
Pass  one  more  from  your  right  hand  to  your  left.     How 
many  in  each  hand  now?     How  many  in  both  hands? 
How  many  are  five  and  one  ? 

5.  Name  the  numbers  in  each  group. 

ft          1  ft  ft      2  ft  ft  ft  3 

ft  ft  ft  ft  ft  _5^  ft  ft  ft  ft  j4_  ft  ft  ft  j$_ 

6  66 

6.  Take  5  objects  and  show  that  5  is  made  up  of  4  and 
1 ;  2  and  3. 

7.  Take  6  objects  and  show  that  6  is  made  of  5  and  1 ; 
4  and  2 ;  3  and  3. 

8.  Take  4  objects  and  show  how  many  ones  are  in  4. 
How  many  twos. 

To  THE  TEACHER. — These  exercises  may 
be  made  very  interesting,  but  they  must  be 
varied  in  order  to  hold  attention. 

Ample  preparation  should  be  made  for 
each  lesson. 


ELEMENTARY  ARITHMETIC 


1  star 
1  star 


ft  ft     2  stars 


ft       1  star 
£  ft     2  stars 
ft  ft  ft  3  stars 


ft  1  star 

ft  ft  3  stars 

ft  ft 

ft  ft  4  8tars 


ft 

ft  ft 
ft  ft 

ft  ft 


ft  ft  ft 
ft  ft  ft 


1  star 

4  stars 

5  stars 

1  star 

5  stars 

6  stars 


LESSON    V. 
Review. 

1.  How  many  stars  are  one  star  and 
one  star  ?     How  many  are  one  and  one  ? 
How  many  ones  in  two  ?    If  you  take  one 
from  two,  how  many  will  you  have  left  ? 

2.  One   star   and   two  stars   are   how 
many?     How  many  are  two  and  one? 
How  many  ones  in  three  ?     If  you  take 
one  from  three,  how  many  will  you  have 
left?     Two  taken  from  three  will  leave 
how  many  ? 

3.  If  you  add  one  star  to  three  stars 
how  many  will  you  have  ?    One  and  three 
are  how  many  ?     One  from  four  leaves 
how  many  ?     Two  from  four  leaves  how 
many?     Three  from  four?     How  many 
ones  in  four  ?     How  many  twos  ? 

4.  Four  stars  arid  one  star   are  how 
many  stars  ?     How  many  are  4  and  1  ? 
How  many  are  2  and  3  ?    How  many  will 
be  left  if  you  take  1  from  5  ?     2  from  5  ? 

3  from  5  ?     4  from  5  ?     How  many  must 
be  added  to  3  to  make  5  ?     How  many 
must  be  added  to  4  to  make  5  ? 

5.  How  many  are  5  stars  and  1  star? 
How  many  are  5  and  1  ?    How  many  are 

4  and  2?     3  and  3?     Taking  1  from  6, 
how  many  are  left  ?    2  from  6?    3  from  6? 
How  many  ones  in  6  ?    How  many  twos  ? 


ELEMENTARY  ARITHMETIC 


LESSON    VI. 
Seven  and  Combinations. 

1.  How  many  stars  in  the  upper  part  of 
the  square  ?    How  many  in  the  lower  part  ? 

2.  How   many  in    the  whole   square? 
How  many  are  6  and  1  ? 

3.  Take  six  objects  in  your  right  hand 
and  one  in  your  left.     How  many  are  in  both  hands  ? 

4.  Pass  one  object  from  right  hand  to  left  hand.    How 
many  in  each  hand  now?     One  and  one  are  how  many? 
One  from  six  leaves  how  many  ?    Fame  the  number  of 
objects  in  each  hand  now.     How  many  in  both  hands? 
5  and  2  are  how  many  ? 

5.  Again  pass  one  object  from  the  right  hand  to  the 
left.    How  many  in  each  hand  now  ?    How  many  in  both 
hands  ?     How  many  are  4  and  3  ? 

6.  Make  seven  marks  on  the  board.    Erase  three ;  how 
many  are  left  ?   If  3  be  taken  from  7,  how  many  will  be  left  ? 

7.  Make  seven  X's  on  the  board.    Erase  two  of  them, 
and  tell  how  many  remain.     7  less  2  are  how  many  ? 

8.  Take  objects  and  show  how  many  threes  are  in  six. 
Show  how  many  twos  in  six. 

9.  If  you  take  two  threes  from  seven,  how  many  will 
be  left  ?     How  many  are  two  threes  ?     Three  twos  ? 

1O.  Complete  the  following : 

6  and  1  are  — .  1  from  7  leaves  — . 
5  and  2  are  — .  2  from  7  leaves  — . 
4  and  3  are  — .  3  from  7  leaves  — . 


To    THE    TEACHER.  —  Pupils    should 


handle  the  objects. 


ELEMENTARY  ARITHMETIC 


LESSON    VII. 
Eight  and  Combinations. 

1.  Take  from  the  table  one  object 
for  every  star  you  see  in  the  upper 
part  of  the  space.  How  many  have 


you  I 

2.  Take    one   more    object.     How 
many  have  you  now  ?     7  and  1  are  how  many  ? 

3.  Place  8  O's  on  ^e  board.     Draw  a  line  so  as  to 
leave  2  Q'8  on  one  side.     How  many  on  the  other  side 
of  the  line.     2  from  8  leaves  how  many  ? 

4.  How  many  are  6  and  2  ?     7  and  1  ? 

5.  Make  a  square  on  the  board.     Draw  a  line  across 
the  square  and  place  5  X's  in  one  space  and  3  X's  in  the 
other.     How  many  X's  in  the  square  ?     5  and  3  are  how 
many  ? 

6.  Take  4  objects  in  each  hand.    How  many  have  you 
in  both  hands?     4  and  4  are  how  many?     How  many 
fours  make  eight  ? 

7.  With  objects  show  how  many  2's  in  8. 

8.  If  you  had  8  cents,  how  many  apples  could  you 
buy  at  2  cents  apiece  ? 

9.  If  Harry  had  8  marbles  and  gave  his  little  brother 
3,  how  many  had  he  left? 

10.  How  many  are  two  2's  ?    Three  2's  ?    Four  2's  ? 

1 1.  Complete  the  following : 

7  and  1  are  — .  1  from  8  leaves  — . 
6  and  2  are  — .  2  from  8  leaves  — . 
5  and  3  are  — .  3  from  8  leaves  — . 
4  and  4  are  — .  4  from  8  leaves  — . 


ELEMENTARY  ARITHMETIC 


ft 


LESSON    VIII. 
Nine  and  Combinations. 

1.  How  many  are  one  and  eight  ? 

2.  Take  two  objects   in  one   hand 
and  seven  in  the  other.    How  many  in 
both  hands  ?     7  and  2  are  how  many  ? 

3.  Take     9     objects.      Give    your 
teacher  3.     How  many  have  you  ?     3  from  9  leaves  how 
many  ?     6  and  3  are  how  many  ? 

4.  Take  9  objects.     Give  your  teacher  4.     How  many 
have  you  ?     How  many  are  5  and  4  ?     Taking  4  from  9, 
how  many  are  left  ? 

5.  Take  9  objects.     Give  your  teacher  3  and  give  one 
of  your  classmates  3.    How  many  have  you  ?    How  many 
threes  in  nine  ? 

6.  John  had  5  rabbits  and  bought  4  more.    How  many 
did  he  then  have  ? 

7.  Maud  cut  9  roses  and  gave  3  to  Irene.     How  many 
did  she  keep  ? 

8.  Take  objects  and  show  that  4  and  4  and  1  are  9. 

9.  Complete  the  following : 

1  and  8  are  — . 

2  and  7  are  — . 

3  and  6  are  — . 

4  and  5  are  — . 

3  and  3  and  3  are  — . 
To  THE  TEACHER. — Place  on  the  black- 
board stars  or  other  simple  figures,  arranged 
as  at  the  top  of  this  page,  and  by  drawing 
vertical  lines  lead  pupils  to  see  that  4  +  4  + 
1  —  9;  2  +  2  +  2  +  2  +  1  =  9.     Use  ob- 
jects freely  and  vary  the  questions. 


1  from  9  leaves  — . 

2  from  9  leaves  — . 

3  from  9  leaves  — . 

4  from  9  leaves  — . 


ELEMENTARY  ARITHMETIC  9 

LESSON    IX. 

"We  have  learned  nine  numbers.     Here  are  the  figures 
which  we  may  use  when  we  want  to  say 

one,  two,  three,  four,  five,  six,  seven,  eight,  nine. 

I     .2    3    M-  .5    lo 

Copy  these  figures  and  try  to  make  them  correctly. 

Here  are  all  the  combinations  you  have  learned.     Now 
try  to  name  the  sum  of  each  one. 


2 
1 

3 

1 

4 
1 

5      I 
1-      1 

1 

1 

8 

1 

23451^ 
222222 

3     4     5     t  •  45 

JL    1    1     3.  .1    .4 

The  pupils  will  name  all  the  combinations  that  make  9. 
All  that  make  8,  7,  6,  5,  4. 

To  THE  TEACHER.  —  Drill  pupils  on  the  table  until  they  can  name  the 
sum  of  each  combination  at  sight.  Then  have  them  name  the  differences. 
And,  lastly,  require  them  to  name  the  sum  of  each  combination  arid  sub- 
tract each  number  in  the  combination  from  that  sum.  Teach  numbers, 
not  figures.  Use  objects  in  your  explanations.  This  table  should  be  placed 
upon  the  board  and  used  for  daily  drills. 


10  ELEMENTARY  ARITHMETIC 

LESSON     X. 
Sign  of  Addition. 

1.  When  we  want  to  show  that  numbers  are  to  be 
added  we  may  use  a  cross  like  this  +. 

2.  It  is  called  "  Plus"  or  the  Sign  of  Addition. 

3.  When  we  find  it  between  numbers  it  is  read  "plus" 
or  "  and."     4  +  2,  is  read  "  four  plus  two"  or  "  four  and 
two." 

4.  Read  these  and  give  the  sum  of  each : 


1. 

3  - 

h  2. 

7. 

4H 

L  3. 

j#. 

5H 

-  2. 

2. 

5  - 

h  3. 

8. 

7  H 

-  1. 

#. 

3H 

h  3. 

3. 

4- 

h  2. 

9. 

5  - 

K4. 

U. 

5- 

h  1. 

4>. 

6  - 

h  1. 

10. 

8  H 

h  1. 

16. 

4n 

h  4. 

5. 

2  - 

h  2. 

11. 

6  - 

h  2. 

17. 

2  - 

h  1. 

6. 

7  - 

r  2. 

12. 

3  - 

h  6. 

18. 

1  H 

h  3. 

To  THE  TEACHER. — Do  not  permit  pupils  to  count.  If  the  sums  can- 
not be  stated  promptly,  use  objects  and  endeavor  to  get  pupils  to  think  of 
the  number  of  objects  represented  by  any  figure. 


LESSON    XI. 
Problems  in  Addition. 

1.  Willie  had  5  cents  and  earned  3  cents.     How  many 
cents  had  he  then  ? 

Answer. — He  had  5  cents  +  3  cents,  or  8  cents. 

2.  Mabel  earned  5  cents  on  Monday  and  2  cents  on 
Tuesday.    How  many  cents  did  she  earn  in  the  two  days  ? 


ELEMENTARY  ARITHMETIC  11 

3.  James  paid  4  cents  for  paper  and  5  cents  for  a 
pencil.     How  many  cents  did  he  pay  for  both  ? 

4.  Charles  had  6  pears  and  "William  gave  him  2  more. 
How  many  had  he  then  ? 

5.  Harry  bought  a  top  for  4  cents  and  a  book  for  4 
cents.     How  much  did  he  spend  ? 

6.  If  a  sheep  costs  6  dollars  and  a  pig  costs  3  dollars, 
how  much  do  both  cost  ? 

7.  Emma  cut  5  roses  from  one  bush  and  2  from  an- 
other.    How  many  did  she  cut  from  both  bushes  ? 

8.  Mary  has  3  daisies  and  Jane  has  2.     How  many 
have  both  ? 

9.  How  many  tops  have  James  and  John  if  each  has 
four  ?    How  many  4's  in  eight  ? 

10.  Maud  has  2  needles  and  May  has  4.     How  many 
have  both  ? 

11.  Arthur  has  4  horses  and  his  cousin  has  3.     How 
many  horses  have  both  ? 

12.  One  ball  team  made  5  runs  and  the  other  made  3 
runs.     How  many  did  both  make  ? 

13.  John    gathered    2    quarts   of  berries    and    Harry 
gathered  3.     How  many  did  both  gather  ? 

14.  How  many  are : 


/. 

4H 

h  2- 

h  1? 

P. 

3 

+  3- 

h2? 

9. 

5  H 

h  2  - 

h2? 

JIA 

4 

+  4- 

h  1? 

3. 

2  - 

h  1  - 

r-  5? 

11. 

5 

+  1  H 

h  3? 

4. 

3H 

-  1  - 

h2? 

12. 

4 

+  2H 

h  3? 

5. 

6  H 

h  1  - 

h  2? 

13. 

3 

+  2H 

h4? 

6. 

1  H 

h  1  - 

h.4? 

14. 

4 

+  IH 

h  3? 

7. 

2  H 

h2- 

h4? 

15. 

6 

+  1  H 

h  1? 

8. 

7  H 

-  1  -i 

h  1? 

16. 

2 

+  2H 

-  1? 

12  ELEMENTARY  ARITHMETIC 

LESSON    XII. 
Sign  of  Subtraction. 

1.  If  we  want  to  show  tliat  one  number  is  to  be  taken 
from  another  we  may  use  a  short  line  like  this  — . 

2.  It  is  called  "  Minus"  or  the  Sign  of  Subtraction. 

3.  When  we  find  this  sign  between  numbers,  it  means 
that  the  number  on  the  right  is  to  be  taken  from  the 
number  on  the  left.     8  —  3,  means  that  3  must  be  taken 
from  8. 

4.  This  sign  is  read  "  minus"  or  "  less."     7  —  4,  is  read 
"  Seven  minus  four,"  or  "  Seven  less  four." 

5.  Eead  the  following  and  name  the  remainders : 


1.   4  —  2. 

6.    6  —  3. 

11.    9  —  5. 

&    7  —  5. 

7.   8  —  4. 

12.    5  —  2. 

3.    8  —  3. 

8.   4  —  3. 

13.    7  —  3. 

4.    6  —  4. 

9.    6  —  2. 

14.    9  —  6. 

5.   7  —  2. 

10.    7  —  4. 

15.   8  —  6. 

6.  Read,  and  name  results : 

^.4  +  4  —  2.  £.8  +  1—4  +  2. 

#.6  —  4+5.  7.    7  —  2  +  4  —  5. 

#.8  —  6  +  3.  £.6  +  2  +  1  —  7. 

4.  7  +  2  —  6.  9.   5  +  2  +  2  —  3. 

5.  5  +  4  —  6.  ./0.    7  —  4  +  3  +  3. 

7.  Take  9  objects  and  show  that  there  are  four  2's  and 
a  1  in  9. 

8.  Show  that  there  are  three  2's  and  a  1  in  7. 

9.  Show  that  there  are  three  3's  in  9. 

To  THE  TEACHER. — Drill  pupils  on  these  and  similar  exercises  until 
they  can  add  and  subtract  rapidly.  At  first  permit  pupils  to  see  the  com- 
binations, and  afterwards  read  them  to  the  class  and  ask  for  results. 


ELEMENTARY  ARITHMETIC  13 

LESSON    XIII. 
Problems  in  Subtraction. 

1.  Oscar  had  9  marbles  and  gave  4  to  his  brother. 
How  many  had  he  left  ? 

Answer. — He  had  9  marbles  —  4  marbles,  or  5  marbles. 

2.  Harvey  started  to  walk  8  miles.     After  he  had 
walked  5  miles,  how  far  had  he  to  go  ? 

3.  Fannie  had  7  roses  and  gave  her  sister  3.     How 
many  did  Fannie  then  have  ? 

4.  Jacob  is  9  years  old  and  his  cousin  is  6  years  old. 
How  much  older  is  Jacob  than  Jiis  cousin  ? 

5.  Laura  was  given  8  words  to  spell.     She  missed  2. 
How  many  did  she  spell  correctly  ? 

6.  James  had  9  cents  and  spent  3  cents  for  a  top. 
How  many  cents  did  he  then  have  ? 

7.  If  you  had  6  dollars  and  spent  3  dollars  for  a  hat, 
how  much  money  would  you  have  left  ? 

8.  George  earned  5  dollars  and  spent  3  dollars.    How 
many  dollars  had  he  left  ? 

9.  There  are  7  pears  in  a  basket.     How  many  will  be 
left,  if  4  are  taken  out? 

10.  Mabel  had  5  cents  and  her  brother  gave  her  4  more. 
If  she  then  spent  4  cents,  how  many  did  she  have  ? 

11.  How  many  are  : 

1.   4  +  5  —  3  +  2  —  3?  tf.6  —  3  +  2  +  4  —  3? 

#.6  —  2  +  5  —  7  —  3?  7.   4  +  4  — 2  +  1  — 5? 

#.5  +  3  —  2  +  3  —  4?  S.   3  +  2  +  2  +  2  — 1? 

£3  +  3  +  3_ 5  +  2?  9.    1  +  8  —  3+1  +  2? 

5.   7  +  2  —  1  —  4  +  3?  10.   3  +  2  —  1+3  —  7? 


14  ELEMENTARY  ARITHMETIC 

LESSON   XIV. 
The  Number  Ten. 

1.  How  many  numbers  have  you  learned  ? 

2.  Write  all  these  numbers  and  tell  how  many  figures 
you  use  in  writing  each  number. 

3.  "What  is  the  largest  number  that  can  be  written  with 
one  figure  ? 

4.  There  is  another  figure  to  learn.    It  is  called  Naught, 
and  is  made  much  like  one  of  the  letters  of  the  alphabet. 
Here  is  the  new  figure  0.     Do  not  forget  its  name. 

5.  Naught  does  not  stand  for  any  number,  but  it  is  of 
use  in  writing  numbers  larger  than  9. 

6.  Nine  and  one  are  ten.     To  write  ten  we 
must  use  naught,  thus  : 

7.  How    many    signs    have   you    learned? 
Make  them  and  tell  what  they  mean. 

8.  Before  going  farther  we  must  learn  another  sign.    It 
is  called  the  Sign  of  Equality,  and  is  made  thus  =.    It  is 
read  "  Equals,"  and  it  means  that  what  stands  on  one  side 
of.  it  has  the  same  value  as  that  which  stands  on  the  other 
side.     For  example,  4  +  6  =  10,  is  read  "  Four'  plus  six 
equals  ten,"  and  means  that  the  sum  of  4  and  6  is  10. 

9.  Headland  complete : 

,?.   3  +  7  =      £  '  7  +  2  =      7.    8  —  3  =        10.   9  — 3  = 

2.   2  +  8=      5.    8  —  5  =      8.   10  —  4=      11.   5  +  4  = 
£9  —  6=      £.6  +  2=      p.    7  +  3=        ^.8  —  2  = 

To  THE  TEACHER. — These  exercises  should  be  extended.  Before  pass- 
ing to  the  next  lesson,  review  the  work  gone  over,  and  see  that  pupils 
understand  how  to  add  and  subtract  rapidly  all  numbers  up  to  and  in- 
cluding ten. 


ELEMENTARY  ARITHMETIC  15 

LESSON    XV. 
Rapid  "Work  on  Combinations. 

1.  Read  and  complete  : 

1.  4  —  2  =  7.  3  +  2  =  13.  5  —  3  =  19.  6  —  4  = 

&  5  +  2  =  &  7  —  5  =  /£.  7  —  2  =  00.  6  +  2  = 

5.  1  +  2  =  9.  2  +  2  =  .75.  3  —  1  =  21.  4  —  2  = 

4.  8  —  3  =  10.  5  +  3  =  16.  8  —  5  =  00.  6  +  3  = 

£.6  —  3=  .7.7.7  +  3  =  77.  10  — 5=  0^.10  — 6  = 

0.  10-r3=  .70.  9  — 6=  .7£.  9  — 4  =  0^.  10  —  5  = 

2.  Read  and  complete : 

^.4  +  3  +  2  —  5  =  7.  1  +  5  —  2  +  6  = 

0.   2  +  3  +  4—1=  5.  5  +  2  —  6  +  4  = 

5.   4  —  2  +  3  +  5=  P.  3  +  3  +  3  —  6  = 

.£.8  —  5  +  4  +  2=  10.  7  +  3  —  5  +  4  = 

5.  6  —  2  +  5  +  2=  11.  6  +  4  —  3  +  1  = 

6.  5  +  2  +  2  —  7=  .70.  2  +  2  +  3  —  1  = 

3.  Find  the  sums  of  these  columns  : 


!•) 

(2-) 

(3.) 

(*•) 

(5.) 

(6.) 

(7.) 

2 

4 

6 

7 

8 

3 

4 

3 

3 

1 

.2 

1 

3 

4 

2 

2 

2 

1 

1 

3 

1 

1 

1 

1 

0 

0 

1 

1 

4.  What  is  the  value  of  the  following : 

(1.)            (2.)          (3.)          (4.)          (5.)          (6.)  (7.)          (8.) 

7           10           9           8           7           6  10           9 

__j2       __^     __j>     _^6     —JJ     —3  — _5  —7; 

5.  How  many  are  10  —  7  ?    10  —  8?    10  —  3?    10- 
2?     10  —  4?     10  —  1?     10  —  9? 


16 


ELEMENTARY  ARITHMETIC 


LESSON   XVI. 
Ten  to  Twenty. 

One  ten  and  no  ones 
are  ten ;  written 

One  ten  and  one  are 
eleven;  written 

One  ten  and  two  are 
twelve;  written 

One  ten  and  three 
are  thirteen;  written 

One   ten    and    four 
are  fourteen ;  written 

One  ten  and  five  are 
fifteen;  written 

One  ten  and  six  are 
sixteen;  written 

One  ten  and  seven 
are  seventeen;  written 

One  ten   and  eight 
are  eighteen ;  written 

One  ten    and    nine 
are  nineteen ;  written 


10 
I  I 


13 


15 
lit 

n 

IS 


ELEMENTARY  ARITHMETIC 


17 


LESSON    XVII. 
The  Number  Eleven. 


10  and  1. 


9  and  2. 


8  and  3. 


7  and  4. 


6  and  5. 


1.  Take  objects  and  show  how  many  are  6  and  5;  7 
and  4 ;  8  and  3 ;  9  and  2. 

2.  Tell  how  to  write  eleven. 

3.  What  does  the  one  on  the  right  stand  for  ? 

4.  "What  does  the  one  on  the  left  stand  for  ? 

5.  If  Maud  had  11  cents  and  spent  3  cents  for  apples, 
how  many  cents  had  she  left  ? 

6.  Rachel  had  11  roses  in  a  bouquet  and  gave  5  to  a 
sick  girl.     How  many  did  Rachel  then  have  ? 

7.  Oscar  earned  11  dollars.     If  he  gave  his  mother  4 
dollars  and  spent  the  remainder  for  a  suit,  how  much  did 
the  suit  cost  ? 

8.  A  man  having  11  pigs  sold  6.     How  many  did  he 
keep? 

9.  Complete  the  following : 


1.  3  +  8  = 

6.  11  —  5  = 

11.  11  —  1  = 

9.  7  +  4  = 

7.  11  —  4  = 

12.  11  —  10  = 

3.  11  —  3  = 

8.  11  —  9  = 

13.  5  +  6  = 

4.  11  -  6  = 

9.  10  +  1  = 

14.  11  —  7  = 

6.  9  +  2  = 

10.  11  —  8  = 

15.  11  —  2  = 

18 


ELEMENTAKY  ARITHMETIC 


LESSON    XVIII. 
The  Number  Twelve. 


10  and  2. 


9  and  3. 


8  and  4. 


7  and  5. 


6  and  6. 


1.  Take  twelve  objects  and  show  that  the  number  12  is 
made  up  of  6  and  6,  of  7  and  5,  of  3  and  9,  of  4  and  8, 
of  10  and  2. 

2.  "Write  twelve  and  tell  what  each  figure  stands  for. 

3.  Bertha  wants  to  buy  a  book  for  twelve  cents.     If 
she  has  a  dime,  how  much  does  she  lack  ? 

4.  Ned  earned  8  cents  on  Monday  and  4  cents  on  Tues- 
day.    How  much  did  he  earn  in  the  two  days  ? 

5.  If  Ned  spent  5  cents  on  Wednesday,  how  much  had 
he  then  ? 

6.  Edna  has  5  pins  in  one  cushion  and  7  in  another. 
How  many  pins  in  both  cushions  ? 

7.  Elmer  put  8  sheep  in  one  field  and  4  in  another  field. 
How  many  in  both  fields  ? 

8.  Ruth  is  6  years  old  and  Albert  is  12  years  old.    How 
much  older  is  Albert  than  Ruth  ? 

9.  Complete  the  following  : 

.7.    12  — 2  =        5.    12  —  6=  9.    12  —  4  = 

&    12  —  3=         6.    12  —  10=         10.   12  —  8  = 

3.  12  —  9  =         7.    10  +  2  =  11.    9  +  3  == 

4.  7  -f  5  =          8.   6  +  6  =  18.    11  +  1  = 


ELEMENTARY  ARITHMETIC  19 

LESSON    XIX. 
Numbers  13,  14,  and  15. 

1.  Show  that  thirteen  is  made  up  of  10  and  3,  9  and  4, 
8  and  5,  7  and  6. 

2.  How  many  are : 

1.  10  +  3?  5.   8  +  5?  9.  11  +  2?  13.  12  +  1? 

&  6  +  7  ?  £.    9  +  4  ?  *0.  13  —  2  ?  1£  13  —  1  ? 

5.  13  —  7?  7.    13  —  9?  £f.  13  —  6?  #.  13  —  5? 

4.  13  —  4?  <?.    13  —  3?  12.  13  —  2?  Jtt.  13  —  10? 

3.  What  does  each  figure  in  thirteen  stand  for  ? 

4.  Using  objects,  show  that  fourteen  is  made  up  of  10 
and  4,  8  and  6,  12  and  2,  9  and  5,  7  and  7,  13  and  1. 

5.  How  many  are  : 

1.  10  +  4?  5.   14  —  4?  9.   14  —  10?  13.    11+3? 

&  9  +  5?  5.    14  —  5?  j?0.    14  —  9?  /£.    12  +  2? 

5.  8  +  6?  7.    14  — 6?  ^.14  —  8?  #.13  +  1? 
£  7  +  7?  5.    14  —  7?  IS.   14  —  3?  ^.   14  —  2? 

6.  "What  is  the  value  of  each  figure  in  fourteen  ? 

7.  "With  objects  show  that  fifteen  is  composed  of  10 
and  5,  9  and  6,  8  and  7,  12  and  3,  11  and  4,  13  and  2,  14 
and  1. 

8.  Find  the  value  of: 

1.   10  +  5  =  5.   15  —  6  =  9.  15  —  8  = 

8.   9  +  6  =  6.   15  —  7  =.  10.  12  +  3  = 

3.   8  +  7  =  7.    15  —  10  =  11.  11  +  4  = 

£15  —  5=  £.15  —  9=  12.  13  +  1  = 

9.  Do  what  the  signs  indicate  : 

11          15  12          14          15  10          9          8 

-|-3       _4       +3       __7       __6       +5     +6     +7 


20  ELEMENTARY  ARITHMETIC 

LESSON   XX. 
Numbers  16  to  20. 

1.  How  many  tens  in  16  ?     In  17  ?     In  18  ?    In  19  ? 

2.  What  is  the  units  figure  in  16?     In  17?     In  18? 
In  19? 

3.  Take  16  objects  and  show  that  8  and  8  are  16. 
That  9  and  7  are  16.     That  10  and  6  are  16.     That  11 
and  5  are  16,  etc. 

4.  In  like  manner  show  the  combinations  of  two  fig- 
ures that  make  17. 

5.  Show  the  combinations  that  make  18. 

6.  Show  the  combinations  that  make  19. 

7.  If  you  were  to  take  the  figure  6  out  of  16,  what 
•figure  should  you  put  in  its  place  ?     Why  ? 

8.  If  you  take  the  figure  1  out  of  the  numbers  16,  17, 
18,  and  19,  what  remainders  would  you  have  ? 

9.  How  much  less  do  you  make  16,  17,  18,  and  19, 
when  you  take  the  1  away.     Why  ? 

1O.  Rapid  work: 

./.  16  — 8  =  10.   16  +  2=  19.  18  —  7  = 

8.  16  —  9  =  11.   17  —  10  =  80.  18  —  6  = 

3.  16  +  3  =  '  18.17  —  1=  81.  18  —  5  = 

4.  16  —  10=  13.    17  —  6=  £#.  19  — 8  = 

5.  16  —  7  =  14.   17  —  5  =  83.  19  —  9  = 

6.  16  —  6  =  15.    18  —  8  =  84.  16  +  0  = 

7.  16  —  5  =  16.   18  —  9  =  85.  19  —  10  = 

8.  17  —  8  =  17.   16  +  1  =  86.  19  —  7  = 

9.  17  —  9  =  18.    18  —  10  =  87.  19  —  6  = 
NOTE. — This  exercise  should  be  extended  by  the  teacher.     "Hasten 

slowly,"  should  be  the  motto. 


1 

2 

3 

4 

1 

1 

1 

1 

2 

3 

4 

5 

1 

1 

1 

1 

3 

4 

5 

6 

1 

3_ 

3_ 

3. 

4 

5 

6 

7 

4 

4 

4 

4 

5 

6 

7 

8 

1 

1 

_5 

5. 

6 

7 

8 

9 

1 

_6 

6. 

_6 

7 

8 

9 

.1 

1 

1 

8 

9 

1 

_8 

9 

9 

ELEMENTARY  ARITHMETIC  21 

LESSON    XXI. 
Table  of  Simple  Combinations. 

56789 
1  ±  i  1  1 

6789 

1111 

789 
1  1  1 

8   9 
4   4 

9 
5 


To  THE  TEACHER.— The  foregoing  table  shows  all  the  combinations 
that  can  be  made  with  the  nine  digits.  Pupils  should  be  drilled  upon  this 
table  until  it  is  thoroughly  learned. 


22  ELEMENTARY  ARITHMETIC 

LESSON    XXII. 

From  Lesson  XXI.  we  derive  the  following : 
Subtraction  Table. 


2 

3 

3 

4    5 

5 

5 

5 

6 

1 

1 

2 

2    1 

2 

3 

4 

1 

6 

6 

6 

6 

7 

7 

7 

7 

7 

7 

2 

3 

4 

5 

1 

2 

3 

4 

5 

6 

8 

8 

8 

8 

8 

8 

8 

9 

9 

9 

1 

2 

3 

4 

5 

6 

7 

1 

2 

3 

9 

9 

9 

9 

9 

10 

10 

10 

10 

10 

4 

5 

6 

7 

8 

1 

2 

3 

4 

5 

10 

10 

10 

10 

11 

11 

11 

11 

11 

11 

6. 

7 

8 

9 

2 

3 

4 

5 

6 

7 

11 

11 

12 

12 

12 

12 

12 

12 

12 

13 

8 

9 

3 

4 

5 

6 

7 

8 

9 

4 

i 

13 

13 

13 

13 

13 

14 

14 

14 

14 

14 

5 

6 

7 

8 

9 

5 

6 

7 

8 

9 

15 

15 

15 

15 

16 

16 

16 

17 

17 

18 

6 

7 

8 

9 

7 

8 

9 

8 

9 

9 

To  THE  TEACHER. — This  table  should  be  learned  thoroughly.     Lead 
pupils  to  see  that,  if  7  and  6  are  13,  13  —  6  —  7,  and  13  —  7  =  6. 


ELEMENTARY   ARITHMETIC 


23 


LESSON    XXIII. 
Twenty  to  Thirty. 


*   ft 

ft   ft 


ft  ft 


10 


9  =  19. 


10 


10  =  20. 


1.  In  19  what  does  1  stand  for  ?    What  does  9  stand  for? 

2.  If  you  increase  19  by  1,  do  you  add  1  to  the  1  or  to 
the  9? 

3.  Adding  1  to  9  will  make  how  many  ?     Adding  1  to 
19,  then,  will  make  how  many  10's? 

4.  In  20  what  figure  shows  that  there  are  two  10's  in 
twenty  ?     Why  is  naught  used  ? 

5.  Count  twenty  objects  and  show  that  there  are  two 
10's  in  20.     Four  5's.     Five  4's. 

6.  How  many  are  2  times  10  ?    4  times  5  ?    5  times  4  ? 

7.  2  tens  and  1  =  twenty-one,  written        2-  I 
2  tens  and  2  =  twenty-two,  written 

2  tens  and  3  =  twenty-three,  written 
2  tens  and  4  =  twenty-four,  written 
2  tens  and  5  =  twenty-five,  written 
2  tens  and  6  =  twenty-six,  written 
2  tens  and  7  =  twenty-seven,  written 
2  tens  and  8  =  twenty-eight,  written 
2  tens  and  9  =  twenty-nine,  written 


J25 


24  ELEMENTARY  ARITHMETIC 

LESSON    XXIV. 

Thirty  to  One  Hundred. 

1.  Three  tens  are  thirty,  written  30 

2.  Write  the  numbers  from  thirty  to  thirty-nine. 

3.  Four  tens  are  forty,  written  40 

4.  Write  the  numbers  from  forty  to  forty-nine. 

5.  Five  tens  are  fifty,  written  50 

6.  Write  the  numbers  from  fifty  to  fifty-nine. 

7.  Six  tens  are  sixty,  written  bO 

8.  Write  the  numbers  from  sixty  to  sixty-nine. 

9.  Seven  tens  are  seventy,  written  HO 

10.  Write  the  numbers  from  seventy  to  seventy-nine. 

11.  Eight  tens  are  eighty,  written  SO 

12.  Write  the  numbers  from  eighty  to  eighty-nine. 

13.  Nine  tens  are  ninety,  written 

14.  Write  the  numbers  from  ninety  to  ninety-nine. 

15.  If  one  be  added  to  ninety-nine,  we  shall  have 

ten  10's  or  one  hundred,  written  I  00 

16.  How  many  tens  in  50?     In  30?     In  40?     In  60? 
In  80?     In  90?     In  70?     In  100? 

17.  How  many  are: 


1. 

70 

+ 

5? 

8. 

22 

+ 

2? 

15. 

92 

+ 

4? 

2. 

60 

+ 

8? 

9. 

36 

+ 

2? 

16. 

75 

+ 

5? 

3. 

40 

+ 

7? 

10. 

41 

+ 

8? 

17. 

28 

+ 

2? 

4- 

90 

+ 

7? 

11. 

62 

+ 

7? 

18. 

65 

+ 

5? 

5. 

50 

+ 

3? 

12. 

56 

+ 

4? 

19. 

83 

+ 

4? 

6. 

30 

+ 

6? 

13. 

72 

+ 

6? 

20. 

36 

+ 

5? 

7. 

20 

+ 

9? 

U. 

84 

+ 

4? 

21. 

90 

+ 

9? 

NOTE.  —  Numbers  from  100  to  1000  should  be  taught  in  succeeding  lessons. 


ELEMENTARY  ARITHMETIC  25 

LESSON    XXV. 

Multiplication. 

1.  If  you  should  want  to  know  how  many  cents  5 
apples  would  cost  at  2  cents  apiece,  you  might  say  2  +  2 
_|_  2  -f-  2  +  2  =  10.     But  there  is  a  shorter  way.     If  we 
add  2  and  2  and  find  the  sum  to  be  4,  or  if  we  add  2 
and  2  and  2  and  find  the  sum  to  be  6,  and  so  on,  we  can 
write  down  these  results  in  the  form  of  a  table  and  then 
commit  the  table  to  memory.     In  the  table  the  sign  X  is 
read  "  times." 

2x1  =  2.  2X4  =  8.  2X7  =  14. 

2X2  =  4.  2  X  5  =  10.        "  2  X  8  =  16. 

2x3  =  6.  2  X  6  =  12.  2  x  9  =  18. 

2.  By  using  objects  prove  that  each  result  in  the  table 
is  correct. 

3.  If  Harry  bought  2  books  at  9  cents  apiece,  how 
much  did  he  spend  ? 

4.  What  will  2  oranges  cost  if  one  orange  is  worth  5 
cents  ? 

5.  At  8  cents  a  quart,  what  will  2  quarts  of  berries  cost  ? 

6.  Edna  bought  2  yards  of  crash  at  8   cents  a  yard. 
How  much  did  she  spend  ? 

7.  William  walked  7  miles  a  day  for  2  days.     How  far 
did  he  walk  ? 

8.  Rapid  work : 

JT.  2X1  +  2=  J.  4  +  8  —  6=  P.  2x8  +  4  = 

£.2x4  —  2=  6.  5  +  7  +  3  =  ^0.  2x4  +  8  = 

#.  2x5  +  5=  7.  7  +  8  —  5=  11.  2  X  7  —  4  = 

£  2X6  —  2=  8.  9  +  7  —  3  =  ./&  2  x  9  +  6  = 


26  ELEMENTARY  ARITHMETIC 

LESSON    XXVI. 
Multiplication  by  Three. 

3x1=3.  3  X  4  =  12.  3  X  7  =  21. 

3X2  =  6.  3  X  5  ==  15.  3  X  8  =  24. 

3X3  =  9.  3X6  =  18.  3x9  =  27. 

1.  How  many  miles  will  a  horse  trot  in  3  hours  if  he 
can  trot  5  miles  in  one  hour  ? 

2.  At  6  cents  a  yard,  what  will  3  yards  of  ribbon  cost  ? 

3.  What  will  9  pencils  cost  at  3  cents  apiece  ? 

4.  Jane  is  7  years  old  and  her  sister  is  3  times  as  old. 
How  old  is  her  sister  ? 

5.  If  one  orange  costs  4  cents,  what  is  the  cost  of  3 
oranges  ? 

6.  Edith  has  3  rose-bushes.     If  she  cuts  8  roses  from 
each  bush,  how  many  roses  will  she  have  ? 

7.  If  a  boy  walks  3  miles  an  hour,  how  far  can  he  walk 
in  9  hours  ? 

8.  Rapid  work : 

1.  8  +  3.  18  +  3.  28  +  3.  38  +  3. 

0.  9  +  5.  19  +  5.  29  +  5.  39  +  5. 

&  8  +  7.  18  +  7.  28  +  7.  38  +  7. 

£6  +  5.  16  +  5.  26  +  5.  36  +  5. 

5.  5  +  8.  15  +  8.  25  +  8.  35  +  8. 

6.  13  —  8.  23  —  8.  33  —  8.  43  —  8. 

7.  15  —  9.-  25  —  9.  35  —  9.  45  —  9. 

8.  14  —  5.  24  —  5.  34  —  5.  44  —  5. 

9.  11  —  7.  21  —  7.  31  —  7.  41  —  7. 

10.  16  —  8.  26  —  8.  36  —  8.  46  —  8. 

11.  3  X  9-  3X8.  3X6.  3X7. 


ELEMENTARY  ARITHMETIC  27 

LESSON    XXVII. 

Division  by  Two  and  Three. 

1.  2  +  2  are  how  many  ?     2  times  2  are  how  many  ? 
How  many  2's  in  4  ? 

2.  If  we  take  6  objects  and  arrange  them  in  three 
parts,  2  in  each,  we  are  dividing. 

3.  Divide  8  into  2  equal  parts.     This  is  dividing  by  2. 
Divide  8  by  4.     Did  you  find  how  many  4's  in  8  ? 

4.  How  many  3's  in  9  ?    With  objects  show  that  your 
answer  is  correct  ? 

5.  Since  3  times  4  =  12,  how  many  4's  in  12  ?     How 
many  3's  ? 

6.  Turn  to  Lesson  XXV.,  examine  the  Multiplication 
Table,  and  tell  how  many  2's  in  4. 

7.  How  many  2's  in  12?     In  16?     In  14?     In  18? 
In  10?     In  8?     In  6? 

8.  To  indicate  Division  we  use  this  sign  -;-.    It  is  read 
"  divided  by." 

9.  Complete  the  following : 

1.    18  -*-  2  =  £   4  -r-  2  =  7.  14  -r-  2  = 

8.    12  -*-  2  ==  5.    6  -r-  2  =  A  16  -s-  2  = 

3.    10  -*-  2  =  0.    2  -*-  2  ==  0.  8  -W  2  = 
1O.  Examine  the  table  in  Lesson  XXVI.,  and  complete 
the  following : 

1.    6  -s-  3  =  £    15  -*-  3  =  7.  18  -s-  3  = 

&    9  -s-  3  =  5.    12  -*-  3  =  8.  27  -5-  3  = 

A   3  -s-  3  =  5.    21  -s-  3  =  P.  24  -5-  3  — 

To  THE  TEACHER. — Lead  the  pupils  to  understand  Division  by  direct- 
ing attention  to  the  fact  that  the  expression  3  X  8  =  24  shows  not  only 
the  product  of  3  and  8,  but  it  shows  how  many  8's  and  how  many  3's  in  24. 


28  ELEMENTARY  ARITHMETIC 

LESSON    XXVIII. 

Multiplication  and  Division. 

1.  With  objects   prove  the  results  in  the  following 
table  and  then  commit  the  table  : 


4X1=4. 
4X2  =  8. 
4x3  =  12. 

4  X  4  =  16. 
4  X  5  =  20. 
4  X  6  =  24. 

4  x  7  =  28. 
4  X  8  =  32. 
4  X  9  =  36. 

2.  How  many  4's  in  8?     In  36?     In  24?     In  20? 
In  16?     In  28?     In  32? 

3.  State  results : 

.7.  4  x  5  —  3.  £  3  X  7  —  4.  7.  12  -*-  3  +  6. 
#.  3  X  6  +  5.  5.  4  X  6  -f  6.  8.  36  -5-  4  +  3. 
&  4  X  9  —  6.  £.9-^-3  +  6.  9.  28  -s-  4  —  2. 

4.  What  will  each  boy  get  if  32  cents  be  divided 
equally  among  4  boys  ? 

5.  What  is  the  cost  of  4  two-cent  postage  stamps  ? 

6.  At  9  cents  a  pound,  what  .will  4  pounds  of  raisins 
cost? 

7.  If  a  man  earns  4  dollars  a  day,  how  much  does  he 
earn  in  4  days  ? 

8.  Divide  24  apples  among  4  boys.    How  many  apples 
will  each  boy  get  ? 

9.  What  is  the  cost  of  4  tops  at  5  cents  apiece  ? 

10.  How  many  eggs  at  4  cents 'apiece  can  be  bought 
for  28  cents  ? 

Slate  Work. 

(1.)  (2.)  (3.)  (4.)  (5.)  (6.) 

2)46        3)36        4)84        2)48        3)96        3)63 


ELEMENTARY  ARITHMETIC 


29 


LESSON    XXIX. 

Multiplication.  Division. 

5X1=  5 ;  therefore,  5-^-5  =  1 
5  X  2  =  10;  therefore,  10  -4-  5  =  2,  and  10  —  2  =  5 
5  x  3  =  15;  therefore,  15  -=-  5  =  3,  and  15  —  3  =  5 
5  X  4  =  20 ;  therefore,  20  -*-  5  ==  4,  and  20  —  4  =  5 
5  x  5  =  25;  therefore,  25  -*•  5  =  5,  and  25  —  5  =  5 
5  X  6  =  30 ;  therefore,  30  -s-  5  =  6,  and  30  —  6  =  5 
5  X  7  =  35 ;  therefore,  35  -f-  5  =  7,  and  35  —  7  =  5 
5  x  8  =  40 ;  therefore,  40  -^  5  =  8,  and  40  —  8  =  5 
5  X  9  =  45 ;  therefore,  45  -5-  5  =  9,  and  45  —  9  =  5 


Rapid  Work. 


.7.   4x4 

--2  +  4 

—  2. 

6.    5 

£.5X4 

-5-2  +  5 

—  3. 

7.    5 

3.   4  X  8 

-r-4  —  4 

+  4. 

8.    5 

4.    5  X  3 

+  4  —  6 

+  7. 

9.   5 

5.   6  X  4 

-=-3  +  7 

—  3. 

10.   5 

Slate  Work. 

(1.) 

(20 

(3.) 

(4-) 

81 

42 

24 

53 

X_5 

X_3 

X  2 

X  3 

(7-)    x 

(8.) 

(9.) 

(10.) 

5)55 

3)69 

5^45^ 

2)86 

(13.) 

(14.) 

(15.) 

3)306 

5)255 

4 

)324 

5X7  —  3  +  6  +  3. 

X9  +  5  +  8  —  6. 

X5  +  5  —  7  —  8. 
5  X  3  +  8  +  7  —  10. 
5  X  8  —  8  +  6  —  10. 


(5.) 

62 

X4 

(ii.) 
4)80 


(16.) 

4 )  288 

: f. i / 

NOTE. — The  teacher  should  explain  with  great  care. 


(6.) 

91 
X_5 

(12.) 
5)50 

(17.) 
3)279 


30  ELEMENTARY  ARITHMETIC 

LESSON    XXX. 
Multiplication  and  Division  by  Six. 

6X1=6.  6  X  4  =  24.  6  X  7  =  42. 

6  X  2  =  12.  6  X  5  =  30.  6  X  8  =  48. 

6  X  3  =  18.  6  X  6  =  36.  6  X  9  =  54. 
From  the  foregoing  table  give  results  to  the  following, 

and  state  reasons  for  results : 

1.    18  -=-  6  =           5.    12  -r-  2  =  9.  54  -f-  6  = 

&    30  -f-  5  =           6.   48  -f-  6  ==  m  30  -5-  6  = 

<?.   36-4-6  =           7.   24-^-6  =  ^.  54 -j- 9  = 

4-   18  -s-  3  =          A   48  -f-  8  =  12.  42  -4-  6  = 

To  THE  TEACHER. — Have  pupils  commit  the  Multiplication  Table,  and 
show  them  that  the  product  of  two  numbers  divided  by  one  of  the  num- 
bers will  give  the  other.  Be  patient ;  do  not  forget  your  own  first  efforts 
to  master  these  tables. 

Remember  that  "Repetition  is  the  mother  of  all  learn- 
ing1." 

LESSON   XXXI. 
Multiplication  and  Division  by  Seven. 

7X1=7.  7  X  4  =  28.          7  X  7  =  49. 

7  X  2  =  14.  7  X  6  =  35.  7  X  8  =  56. 
7  X  3  =  21.           7  X  6  =  42.  7  X  9  =  63. 

Give  results  and  state  reasons,  as  in  the  preceding 
lesson : 


1.   42  -4-  7  = 

-5.   28-4-7  = 

P.    49  -4-  7  = 

2.    21  -f-  7  = 

(5.   63  -f-  7  = 

.70.    56-4-7  = 

5.    14  -4-  7  = 

7.   35  -4-  7  = 

11.   42  4-  6  = 

4.   28-4-4  = 

A    63  -4-  9  =« 

j?&    14  -4-  2  == 

ELEMENTARY  ARITHMETIC  31 

LESSON    XXXII. 
Multiplication  and  Division  by  Bight. 

8X1  =  8.  8  X  4  =  32.  8  X  7  =  56. 

8  X  2  =  16.  8  X  5  =  40.  8  X  8  =  64. 

8  X  3  =  24.  8  X  6  =  48.  8  X  9  =  72. 

Complete  the  following  as  in  former  lessons : 


1. 

40 

*-8  = 

6.   24 

-s-8 

= 

.77. 

16 

-j- 

8 

= 

2. 

72 

-5-8  = 

7.    16 

-^-2 

r= 

/& 

64 

•4- 

8 

= 

3. 

48 

-4-  6  = 

8.    64 

-4-8 

= 

13. 

40 

-^ 

5 

= 

5. 

32 
56 

-4-4  = 

.      *y   

9.   72 
10.   48 

-4-8 

= 

U- 
15. 

32 
56 

-r- 

8 
8 

= 

Slate 

Work. 

(1 

'1 

64_ 

(2.) 

2)48 

(3.) 

3)66 

2 

(4-) 
)40 

(5 
41 

48_ 

(6.) 

2)648 

LESSON    XXXIII. 
Multiplication  and  Division  by  Nine. 

9X1  =  9.  9  X  4  =  36.  9  X  7  =  63. 

9  x  2  =  18.  9  X  5  =  45.  9  x  8  =  72. 

9  X  3  =  27.  9  X  6  =  54.  9  X  9  =  81. 

Complete  the  following : 


1.   54  _:_  9  = 

6.   45  -4-  9  = 

11.   36-4  = 

&   81  -*-  9  = 

7.    27-^9  = 

12.    18-5-9  = 

8.   72  -*-  8  = 

A    18  -f-  2  = 

IS.   54  -5-  6  = 

4.   45  -*-  5  = 

p.    36  -5-  9  = 

14.    72  -5-  9  = 

5.    63-;-  9  = 

10.    64  -r-  8  = 

#.   63-=-  7  = 

32  ELEMENTARY  ARITHMETIC 

LESSON    XXXIV. 
Miscellaneous  Exercises. — Slate  Work. 


1. 

Add: 

PO 

2 

(2-) 

3 

(3.) 

4 

(4.) 

2 

(5.) 
3 

(6.) 

4 

(70 
2 

(8.) 
1 

3 

5 

5 

6 

8 

5 

2 

5 

4 

2 

6 

4 

2 

6 

3 

8 

5 

1 

3 

5 

1 

7 

3 

6 

(9.) 

9 

(10.) 

8 

(11.) 
9 

(12.) 
3 

(130 

10 

(14.) 
12 

(15.) 

14 

(16.) 

19 

3 

6 

8 

9 

3 

2 

6 

1 

2 

3 

3 

4 

5 

1 

5 

7 

1 

2 

1 

6 

4 

8 

5 

2 

2. 

Subtract 

PO 

63 

(2-) 

89 

(3.) 

34 

(4.) 
87 

(6.) 
59 

(6.) 
99 

(7.) 
78 

42 

73 

13 

35 

27 

35 

18 

3. 

Multiply 

: 

PO 

12 

(2-) 

14 

(3.) 

21 

(4.) 
34 

(5.) 
22 

(6.) 
21 

(7-) 
32 

2 

2 

3 

2 

4 

5 

4 

(8.) 
15 

(9.) 
24 

(10.) 
36 

(11.) 
29 

(12.) 

34 

(13.) 

52 

(14.) 
64 

2 

3 

4 

6 

6 

7 

8 

To  THE  TEACHER. — These  simple  exercises  should  be  extended  until 
pupils  acquire  facility  in  performing  the  operations.  CAUTION  :  Do  not 
increase  difficulties  too  rapidly ;  carefully  explain  every  step. 


ELEMENTARY  ARITHMETIC  33 

LESSON    XXXV. 

Common  Fractions. 

HALVES. 

1.  If  you  wanted  to  share  an  apple 
equally  with  your  brother,  into  how  many 
parts  would  you  cut  the  apple  ? 

2.  What  part  of  the  apple  would  your 
brother  get  ?    What  part  would  you  keep  ? 

3.  How  many  halves  in  one  apple  ?    How  many  halves 
in  1? 

4.  How  many  halves  in  2  ?     In  3  ?     In  4  ?     In  5  ? 

5.  To  find  one-half  of  a  number  divide  by  2. 

6.  One-half  is  written  ^-. 

7.  Find  |  of  4,  6,  8,  12,  14,  16,  10,  20,  28. 

8.  Find  £  of  5,  7,  9,  13,  15,  11,  17,  19,  21. 

9.  Divide  16  oranges  between  2  boys. 

10.  If  two  slates  cost  22  cents,  what  is  the  cost  of  one 
slate  ? 

11.  A  boy  had  14  cents,  and  lost  one-half  of  them. 
How  many  had  he  left  ? 

12.  If  you  had  18  cents  and  should  lose  one-half  of 
your  money,  how  many  oranges  at  3  cents  apiece  could 
you  buy  for  the  remainder  ? 

13.  John  and  James  caught  12  fish,  and  divided  them 
equally.     How  many  did  each  get  ? 

14.  If  you  had  1  melon,  to  how  many  persons  could 
you  give  ^  melon  ? 

To  THE  TEACHER. — The  teacher  will  explain  No.  8,  and  by  the  use 
of  objects  both  Nos.  7  and  8. 

3 


34  ELEMENTARY  ARITHMETIC 

LESSON    XXXVI. 
FOURTHS. 

1.  If  a  square  be  divided  into  four 
equal  parts,  what  is  one  part  called  ? 

2.  How  many  fourths  in  anything? 
How  many  fourths  in  1  ? 

3.  How  do  you  write  one-fourth  ? 

4.  What  part  of  one-half  is  one-fourth  ? 

5.  How  many  fourths  in  one-half? 

6.  How  many  fourths  in  one  dollar?     In  2  dollars? 
In  4  dollars  ?     In  5  dollars  ? 

7.  To  find  one-fourth,  into  how  many  parts  do  you 
divide  a  thing  ? 

8.  What  do  you  divide  by  to  find  J  of  a  number  ? 

9.  What  is  J  of  4?     Of  12?     Of  20?     Of  24? 

10.  If  4  boys  share  one  dollar  equally,  what  part  of  the 
dollar  does  each  boy  get  ? 

1 1.  "What  is  one-third  of  3  apples  ?     What  is  2  thirds 
of  3  apples  ?     3  thirds  ? 

12.  If  12  roses  be  equally  divided  among  3  girls,  how 
many  will  each  have  ? 

13.  What  is  J  of  28  ?     Of  32?     Of  36?     Of  16? 

14.  Take  an  apple  and  show  how  you  would  find  |  of 
an  apple.     How  J  of  an  apple.     Three-fourths. 

15.  What  is  J  of  9?     Of  13?     Of  17?     Of  21  ? 

16.  Find  |  of  4,  8,  12,  16,  20,  24,  28. 

17.  Two-fourths  of  an  orange  is  what  part  of  it? 

To  THE  TEACHER.— 'Make  many  simple  problems  requiring  the  finding 
of  one-fourth. 


ELEMENTARY  ARITHMETIC 

LESSON    XXXVII. 
EIGHTHS. 


35 


*  1  i 

i    |    i 

Ej 

i 

j 

1.  How  many  halves  are  shown  above  ?     How  many 
fourths  ? 

2.  Into  how  many  parts  is  each  fourth  divided? 

3.  Into  how  many  parts  is  the  whole  space  divided  ? 

4.  If  anything  be  divided  into  8  equal  parts,  what  is 
one  part  called  ? 

5.  How  do  you  write  one-eighth  ?     Three-eighths  ? 

6.  How  many  eighths  in  anything  ? 

7.  How  many  eighths  in  one-half  of  anything? 

8.  How  many  eighths  in  one-fourth  of  anything? 

9.  How  do  you  find  one-eighth  of  an  apple  ?     Take 
an  apple  and  show  how  it  is  done. 

10.  How  do  you  find  ^  of  16  objects  ?    Show  how  with 
objects. 

1 1.  How  would  you  find  J  of  any  number  ? 

12.  What  is  |  of  22  ?     Of  40  ?     Of  80  ?     Of  32  ? 

13.  Find  |  of  8,  16,  24,  32,  40,  48,  56. 

14.  How  many  eighths  in  1  ?     In  2,  3,  4,  5,  and  6  ? 

15.  What  is  £  of  17,  25,  41,  49,  57  ? 

16.  What  part  of  a  number  do  you  get  when  you  divide 
by  2?     By  4?     By  8? 

17.  Find  £  of  16,  32,  24,  40,  56,  72,  80,  and  48. 


36  ELEMENTARY  ARITHMETIC 

LESSON    XXXVIII. 
THIRDS. 

1.  Into  how  many  equal  parts  is  the 
square  divided  ? 

2.  What  is  each  part  called  ? 

3.  How  many  thirds  in  one  square  ? 

4.  If   you    should    spend    f   of   your 
money,  how  many  thirds  would  you  have  ? 

5.  How  do  you  find  one-third  of  an  orange  ? 

6.  How  many  thirds  in  1  ?     In  2  ?     In  3  ?     In  4  ? 

7.  How  do  you  find  J  of  a  number  ? 

8.  What  is  one-third  of  6,  9,  15,  18,  12,  24,  30? 

9.  How  do  you  find  two-thirds  of  a  number  ? 

10.  Find  f  of  6,  12,  15,  9,  18,  21,  30,  24,  27. 

11.  If  Edna  is  15  years  old  and  Edith  is  two-thirds  as 
old,  how  old  is  Edith  ? 

12.  Harry  had  21  problems  to  solve.     He  solved  f  of 
the  number.     How  many  did  he  miss  ? 

13.  Walter  started  to  ride  36   miles  on   his  bicycle. 
After  he  had  gone  f  of  the  distance,  how  far  had  he 
to  go? 

14.  A  boy  having  33  cents,  spent  f  of  his  money.    How 
much  did  he  spend  ?     How  much  had  he  left  ? 

15.  What  is  the  difference  between  one-third  and  one- 
fourth  of  24? 

16.  At  12  dollars  a  barrel,  what  will  f  of  a  barrel  of 
fish  cost? 

17.  What  is  J  of  7,  9,  13,  16,  19,  22,  25? 

18.  Albert  has  J  of  27  dollars  and  Roscoe  has  four 
t^imes  as  much.     How  many  dollars  has  Roscoe  ? 


ELEMENTARY  ARITHMETIC 


37 


LESSON    XXXIX. 

SIXTHS. 

1.  If   anything    be    divided    into    six 
equal  parts,  each  part  is  called  one-sixth. 
Written  f 

2.  How  many  thirds  in  the  square  ? 

3.  Into  how  many  parts  is  each  third 
divided  ? 

4.  How  many  sixths   in   the   square? 
How  many  sixths  in  each  third  ? 

5.  Does  ^  =  -|?     \  =  how  many  sixths? 

6.  Draw  a  square  and  show  that  ^  =  f . 

7.  How  do  you  find  one-sixth  of  a  thing? 

8.  How  do  you  find  one-sixth  of  a  number? 

9.  State  ^  of  6,  18,  24,  12,  30,  42,  48,  36. 

10.  How  many  sixths  in  1,  3,  2,  4,  6,  and  5? 

11.  State  f  of  6,  24,  1 2,  18,  30,  36,  42,  and  48. 

12.  Five-sixths  of  24  years  is  twice  the  age  of  Charles. 
How  old  is  Charles? 

13.  Maud  has  ^  as  many  roses  as  Jane.     How  many 
has  Maud  if  Jane  has  30  roses  ? 

14.  If  William  has  -f-  of  18  cents,  how  much  more  does 
he  need  to  buy  10  two-cent  stamps  ? 

15.  If  one  man  earns  42  dollars  and  another  earns  %  as 
much,  how  much  will  the  second  man  earn  ? 

16.  Complete  the  following : 

1.  I  of  60  =  5.  f  of  48  =  9.  f  of  36  = 

&  £  X  5  =  6.  i  X  7  =  10.  |  X  4  = 

S.  |  of  27  =  7.  \  of  84  9  ^.  £  of  96  — 

£  }  X  10  =  &  f  x  12  =  m  f  of  84  = 


38  ELEMENTARY  ARITHMETIC 

LESSON    XL. 

FIFTHS  AND  TENTHS. 

1.  If  a  pear  be  divided  into  5  equal  parts,  each  part  is 
called  one-fifth ;  written  ^. 

2.  What  are  two  parts  called  ?     Three  parts  ?    Four 
parts  ? 

3.  Write  two-fifths,  three-fifths,  four-fifths. 

4.  How  is  one-fifth  of  a  number  found  ?     Two-fifths  ? 
Three-fifths  ?    Four-fifths  ? 

5.  Find  £  of  10,  20,  30,  40,  15,  25,  50,  35,  45. 

6.  Complete  the  following : 

1.  £  of  20  =          4.  |  of  35  =  7.   f  of  25  = 

8.   £  of  40  =          5.   £  of  45  =  8.   £  of  30  = 

8.   f  of  15  =          6.  |  of  50  =  9.   $  of  10  = 

7.  If  anything  be  divided  into  10  equal  parts,  each 
part  is  called  one-tenth ;  written  -fa. 

8.  Write  three-tenths,  five-tenths,  seven-tenths,  nine- 
tenths. 

9.  Can  you  show  that  ^  =  £,  ^  =  fc  A  =  f  ? 

10.  How  many  tenths  in  1,  2,  3,  5,  7,  6,  9,  8  ? 

11.  Name  fa  of  40,  50,  80,  100,  70,  60,  30. 

12.  ^  of  60  are  how  many  times  3? 

13.  A  drover,  having  100  sheep,  sold  ^5-  of  them  to 
one  man  and  fa  to  another.     How  many  did  he  have 
left? 

14.  Complete  the  following : 

1.   -fa  of  30  =  4.   &  of  60  =  7.    &  of  100  = 

&   -ft-  of  80  =  5.   ^  of  20  =  8.   A  of  100  = 

8.   ^  of  70  =  6.   &  of  40  -=  9.   ^  of  100  = 

To  THE  TEACHER.— Simple  problems  should  be  given  here. 


ELEMENTARY  ARITHMETIC  39 

LESSON    XLI. 
General  Review. 

1.  Make  all   the   signs   that  you  have  learned   and 
explain  their  use. 

2.  How  many  processes  have  you  learned  ? 

3.  Make  a  simple  problem  in  Addition  and  solve  it. 

4.  Make  a  Subtraction  problem,  solve  and  explain  it. 

5.  Repeat  the  Multiplication  Table. 

6.  Show  how  Division  may  be  learned  from  the  Mul- 
tiplication Table. 

7.  Tell  what  is  meant  by  one-half.     One-third.     One- 
fourth.    One-fifth.    One-sixth.    One-seventh.    One-eighth. 
One- ninth.     One-tenth. 

8.  Show   how   many   halves,   thirds,   fourths,   fifths, 
sixths,  sevenths,  eighths,  ninths,  and  tenths  in  1. 

9.  How  do  you  find  one-sixth  of  a  number?     One- 
tenth?     One-third?     One-fifth? 

10.  Divide  each  of  the  following  numbers  by  7  and 
by  9  and  name  the  remainders,  if  any:   63,  81,  49,  54, 
90,  42,  38. 

11.  What  parts  of  the  numbers  did  you  find  in  No.  10  ? 

12.  Complete  the  following  : 

1.    2X3  +  5  —  1=  8.  18  -s-  3  +  20  —  13  = 

£.8-^-2  —  2  +  9  =  9.  30  -*-  3  —  10  +  9  = 

3.  9x4-^-3  +  8  =  10.  J-  of  36  X  4  —  9  = 

4.  6x2x2  +  6=  11.  f  of  24 --2  +  8  = 

5.  5X4x4-f-2=  18.  f  of  36  —  9  +  6  = 

6.  18  --  3  +  6  —  5  =  IS.  27  +  3  —  16  -*-  2  = 

7.  of  90  —  1  +  20  =    14.   f  of  42  +  6  +  4  X  2  = 


40  ELEMENTARY  ARITHMETIC 

LESSON  XLII. 
State  results  promptly : 


1. 

1  Q    _]_    Q 
1  f/   ~Y~    V   ^=: 

28. 

45-^-5  = 

55. 

18  -*-  9  = 

2. 

1  of  18  = 

29. 

|  of  45  = 

56. 

27  —  9  = 

3. 

27  —  9  = 

30. 

27  x  i  = 

57. 

37  +  7  = 

4. 

42  -5-  7  = 

31. 

7X7  = 

58. 

18-5-6  = 

5. 

9X8  = 

32. 

f  of  49  = 

59. 

1  of  18  = 

6. 

63-^-9  = 

33. 

8X8  = 

60. 

I  of  18  = 

7. 

49  +  9  = 

34. 

3  X  12  = 

61. 

99  +  9  = 

8. 

f  of  24  = 

35. 

24  —  9  = 

62. 

99-5-9  = 

9. 

f  of  18  = 

36. 

64  —  8  = 

63. 

t  of  99  = 

10. 

9X9  = 

37. 

33  -*-  3  = 

64. 

80  —  10  = 

11. 

90  -f-  10  = 

38. 

£  of  25  = 

65. 

-jV  of  80  = 

12. 

37  —  8  = 

39. 

f  of  25  = 

66. 

6X4  = 

13. 

i  of  36  = 

40. 

6X6  = 

67. 

4x6  = 

14. 

|  of  36  = 

41. 

£of  36  •  = 

68. 

|  of  24  = 

15. 

81—9  = 

42. 

|  of  36  = 

69. 

T3¥  of  40  = 

16. 

76  +  9  = 

43. 

28  +  9  = 

70. 

i  X  42  = 

17. 

21  —  7  = 

44. 

28  -5-  7  = 

71. 

42  —  6  = 

18. 

69  +  9  = 

45. 

36  -6  = 

72. 

59  +  9  = 

19. 

63  —  7  = 

46. 

49  -5-7  = 

73. 

69  +  9  = 

20. 

^  of  90  = 

47. 

72  -5-9  = 

74. 

7X8  = 

21. 

A  of  90  = 

48. 

9X8  = 

75. 

56  -f-  8  = 

22. 

14+19  = 

49. 

21  —  7  = 

76. 

56  -j-  7  = 

23. 

18  —  11  = 

50. 

|  of  21  = 

77. 

|  of  56  = 

24. 

72  —  8  = 

51. 

f  of  21  = 

78. 

|  of  56  = 

25. 

J  of  27  = 

52. 

29  +  8  = 

79. 

9X4  = 

26. 

|  of  36  = 

53. 

37  —  29  = 

80. 

36  —  9  = 

27. 

f  of  18  = 

54. 

f  of  24  = 

81. 

£  of  36  = 

This  page  may  be  divided  into  two  or  more  lessons. 


ELEMENTARY  ARITHMETIC 


PART    II. 


DEFINITIONS. 

1.  A  Unit  is  a  single  thing ;  as,  a  book,  a  man,  an  hour. 

2.  A  Number  is  a  unit,  or  a  collection  of  units;   as, 
one  hat,  ten  birds,  twenty-Jive  dollars. 

3.  An   Abstract    Number   is  one  whose    unit   is   not 
named;  as,  3,  10,  21. 

4.  A  Concrete  Number  is  one  whose  unit  is  named; 
as,  6  pens,  7  days,  10  dollars. 

All  abstract  numbers  have  the  same  unit;  hence,  the 
value  of  an  abstract  number  depends  entirely  upon  the 
number  of  its  units.  Concrete  numbers  do  not  all  have 
units  of  the  same  value,  and  therefore  the  value  of  a  con- 
crete number  depends  both  upon  the  value  of  its  unit 
and  upon  the  times  its  unit  is  repeated.  A  five-dollar  bill 
is  more  valuable  than  a  two-dollar  bill,  for  five  repetitions 
of  the  unit,  one  dollar,  give  'a  greater  value  than  two 
repetitions.  Five  dozen  eggs  is  a  greater  number  than 
five  eggs,  for,  while  the  repetitions  of  the  unit  are  the 
same,  the  unit,  one  dozen  eggs,  is  greater  than  the  unit, 
one  egg.  Hence,  the  Unit  of  a  Number  is  the  measure 
of  the  number,  and  determines  its  value. 

41 


42  ELEMENTARY  ARITHMETIC 

5.  Arithmetic  explains  numbers  and  teaches  methods 
of  using  them. 

6.  In  writing  numbers,  characters  called  Figures  are 
used,  and  also  certain  Capital  Letters. 

7.  The  writing  of  numbers  with  figures  or  with  letters 
is  called  Notation. 

8.  The  reading  of  numbers  is  called  Numeration. 

EXERCISES. 

1.' What  is  a  unit?  How  many  units  are  there  in 
eight  ? 

2.  Compare  eight  pounds  with  eight  pounds.     Have 
these  numbers  the  same  unit?     Has  each  the  same  num- 
ber of  units  ?     Then,  if  you  are  asked  why  eight  pounds 
equal  eight  pounds,  what  answer  can  you  give  ? 

3.  In  like  manner  compare  the  following :  Ten  ounces 
and  six  ounces.     Ten  dozen  eggs  and  ten  eggs.     Twelve 
feet  and  ten  inches. 

4.  Name  the  unit  in  each  of  the  following,  and  tell 
which  are  concrete  and  which  abstract : 

1.  Four.  6.  Twelve  dollars. 

2..  Nine  ships.  7.  Eleven. 

3.  Two  sailors.  8.  Three  eagles. 

4.  Seven.  9.  Twenty. 

5.  Four  guns.  10.  Thirty  horses. 

NOTATION   AND   NUMERATION. 

DEFINITIONS. 

1.  The  system  of  notation  employing  figures  is  called 
the  Arabic  System;  that  employing  letters  is  called  the 
Roman  System. 


NOTATION  AND  NUMERATION  43 

The  one  was  introduced  into  Europe  by  the  Arabs  ;  the  other  was  used 
by  the  ancient  Komans. 

2.  The  Arabic  system  employs  ten  figures  to  represent 
numbers  . 

0,       1,      2,      3,      4,      5,     6,      7,       8,      9. 

Names  :  Naught,  one,    two,  three,  four,   five,   six,   seven,  eight,  nine. 

Naught  is  also  called  zero  and  cipher.  The  rest  are 
called  significant  figures. 

Because  we  have  ten  fingers,  these  ten  figures  are 
sometimes  called  digits.  [Latin,  digitus,  finger.] 

3.  The  following  are  correct  forms  for  these  digits  : 


Slant  Script  Figures. 


Vertical  Script  Figures. 

EXERCISES. 
Make  the  best  Arabic  forms  you  can  for  the  following  : 

1.  Naught,  one,  two,  three,  four,  five,  six,  seven, 

eight. 

2.  Two,  three,  four,  five,  six,  seven,  eight,  nine. 

3.  Three,  four,  five,  six,  seven,  eight,  nine,  naught. 

4.  Four,  five,  six,  seven,  eight,  nine,  naught,  one. 

5.  Five,  six,  seven,  eight,  nine,  naught,  one,  two. 

6.  Six,  seven,  eight,  nine,  naught,  one,  two,  three. 

7.  Seven,  eight,  nine,  naught,  one,  two,  three,  four. 

8.  Eight,  nine,  cipher,  one,  two,  three,  four,  five. 

9.  Nine,  zero,  one,  two,  three,  four,  five,  six. 

10.  Naught,  one,  two,  three,  four,  five,  six,  seven. 


44  ELEMENTARY  ARITHMETIC 

Numbers  from  Ten  to  Twenty. 

1.  The  number  next  above  nine  is  called  ten  (10). 

2.  Notation  forms  numbers  into  groups  of  ten  each. 
The  first  group  is :  1,  2,  3,  4,  5,  6,  7,  8,  9,  10. 

3.  The   second  group  of  ten   numbers   is   formed  as 
follows  : 

1 1 ,  named  eleven,  one  and  ten. 

12,  named  tivelve,  two  and  ten. 

13,  named  thirteen,  three  and  ten. 

14,  named  fourteen,  four  and  ten. 

15,  named  fifteen,  five  and  ten. 

16,  named  sixteen,  six  and  ten. 

17,  named  seventeen,  seven  arid  ten. 

18,  named  eighteen,  eight  and  ten. 

19,  named  nineteen,  nine  and  ten. 

20,  named  twenty,  twice  ten  (ty  means  ten). 

EXERCISES. 

1.  To  be  named: 

1.  11,     15,     20,     13.          6.  16,     13,     12,     18. 

2.  19,     16,     14,     19.         7.  20,     17,     16,     13. 
.    3.  12,     17,     11,     15.         8.  11,     14,     13,     19. 

4.  18,     15,.   19,     16.          9.  18,     16,     17,     14. 

5.  13,     18,     15,     17.        10.  12,     15,     12,     20. 

2.  To  be  expressed  in  figures : 

1.  Ten,  eighteen,  eleven,       fourteen. 

2.  Eleven,  nineteen,  seventeen,  eleven. 

3.  Twelve,  fifteen,      twenty,      eighteen. 

4.  Thirteen,  twelve,      sixteen,      twenty. 

5.  Fourteen,  twelve,      nineteen,   seventeen. 


NOTATION  AND  NUMERATION  45 

Each  of  the  above  numbers  consists  of  two  figures : 
the  figure  on  the  right  expresses  units  ;  the  figure  on  the 
left,  tens. 

3.  Point  out  the  number  of  tens  and  units  in  11,  17, 
19,  15,  20. 

Numbers  from  Twenty  to  One  Hundred. 

The  numbers  from  one  to  one  hundred  form  ten  groups. 
The  first  two  groups  we  have  shown  you. 

Third  group :  21,  22,  23,  etc.,  named  twenty-one, 
twenty- two,  etc. 

Fourth  group:  31,  32,  33,  etc.,  named  thirty-one, 
thirty-two,  etc. 

Fifth  group:  41,  42,  43,  etc.,  named  forty-one,  forty- 
two,  etc. 

Sixth  group:  51,  52,  53,  etc.,  named  fifty-one,  fifty- 
two,  etc. 

Seventh  group :  61,  62,  63,  etc.,  named  sixty-one, 
sixty-two,  etc. 

Eighth  group  :  71,  72,  73,  etc.,  named  seventy-one, 
seventy-two,  etc. 

Ninth  group:  81,  82,  83,  etc.,  named  eighty-one, 
eighty-two,  etc. 

Tenth  group :  91,  92,  93,  etc.,  named  ninety-one, 
ninety-two,  etc. 

NOTE. — The  last  number  of  the  tenth  group  is  called  one  hundred  (100). 

EXERCISES. 

1.  Write  in  figures,  and  read,  the  ten  numbers  of  the 
third  group. 


46 


ELEMENTARY   ARITHMETIC 


2.  Write  in  figures,  and  read,  the  numbers  belonging 
to  each  of  the  remaining  groups. 

3.  Read  each  of  the  following  numbers,  and  name  the 
tens  and  units  in  each  : 

(0.) 
42, 
65, 
87, 
98, 
50, 
48, 
53, 
40, 
77, 
80, 


(A.) 

(B.) 

1. 

64, 

45, 

2. 

73, 

93, 

3. 

92, 

82, 

4. 

18, 

67, 

5. 

30, 

60, 

6. 

75, 

23, 

7. 

26, 

78, 

8. 

38, 

56, 

9. 

44, 

29, 

10. 

20, 

90, 

(D.) 

(E.) 

67, 

49. 

43, 

53. 

79, 

19. 

32, 

24. 

91, 

86. 

28, 

27. 

17, 

60. 

65, 

67. 

25, 

72. 

68, 

83. 

4.  Express  in  figures  the  following : 


(A.) 

1.  One  unit. 

2.  Two  units. 

3.  One    ten   and 

three  units. 

4.  Two  tens  and 

five  units. 

5.  Six    tens    and 

five  units. 

6.  Eight  tens  and 

three  units. 

7.  Seven  tens  and 

six  units. 

8.  Two  tens  and 

nine  units. 


(B.) 

Forty-six. 
Seventy-five. 


(C.) 
Sixty. 
Thirty-eight. 


Twenty-seven. 

Seventy-six. 

Sixty-eight. 


Fifty-three. 
Thirty-four. 
Sixty-nine. 
Ninety-seven.  Thirty-two. 
Eighty.  Eighty-nine. 

Twenty-three.  Thirty-five. 


NOTATION  AND  NUMERATION  47 

(A.)  (B.)  (C.) 

9.  Eight  tens  and 

seven  units.     Eighty-one.      Ninety-four. 
10.  Nine  tens  and 

no  units.  Seventy-two.    Forty-one. 

5.  What  does  the  ty  in  twenty,  thirty,  etc.,  mean? 

6.  Does  seventy  mean  seven  and  ten,  or  seven  tens  ? 

7.  Write  and  read  all  the  numbers  of  the  first  group. 

8.  Write  and   read   all  the  numbers  of  the   second 
group.     W  equals  how  many  tens  f 

9.  Write  and  read  all  the  numbers  of  the  tenth  group. 

10.  How  many  groups  of  ten  numbers  each  have  to  be 
written  before  reaching  one  hundred  (100)  ? 

11.  Then  100  equals  how  many  tens  ?    10  equals  how  many 

units  ? 

Numbers  from  1OO  to  1OOO. 

1.  Since  one  hundred  is  written      100, 

two  hundred  is  written     200, 

three  hundred  is  written  300, 

four  hundred  is  written    400, 

five  hundred  is  written     500, 

six  hundred  is  written       600, 

seven  hundred  is  written  700, 

eight  hundred  is  written  800, 

nine  hundred  is  written    900. 
Nine  is  written  thus,  9, 

9  units. 
Ninety-nine  is  written  thus,  99, 

9  tens  and  9  units. 
Nine  hundred  ninety-nine  is  written  thus,  999, 

9  hundreds,  9  tens,  and  9  unite. 


48  ELEMENTARY  ARITHMETIC 


PRINCIPLE. 

A  figure  standing  alone,  or  in  the  right-hand 
place,  expresses  units;  in  the  second  place,  tens; 
in  the  third  place,  hundreds. 


2.  Analysis  (Greek,  taking  apart)  points  out  the  parts 
of  which  a  number  is  composed. 


EXERCISES. 

1.  Analyze  and  read  the  number  473.     To  apply  the 
principle :   4  standing  in  the  third  place  expresses  hun- 
dreds, 7  standing  in  the  second  place  expresses  tens,  3  in 
the  first  place  expresses  units.    Hence,  the  number  is  read 
four  hundred  seventy-three.     Do  not  read  thus  :  Four  hun- 
dred and  seventy- three.      The  importance  of  dropping 
and  will  appear  later. 

2.  In  like  manner  analyze  and  read  the  following : 


(A.) 

(B.) 

(C.) 

(D.) 

(E.) 

1. 

613, 

702, 

962, 

814, 

104. 

2. 

724, 

359, 

483, 

902, 

134. 

3. 

538, 

400, 

965, 

893, 

246. 

4. 

904, 

916, 

684, 

246, 

408. 

5. 

523, 

820, 

793, 

489, 

325. 

6. 

186, 

547, 

924, 

249, 

767. 

7. 

248, 

962, 

998, 

568, 

482. 

8. 

517, 

483, 

594, 

346, 

618. 

9. 

342, 

965, 

489, 

200, 

821. 

10. 

981, 

684, 

993, 

108, 

604. 

NOTATION  AND  NUMERATION  49 

3.  Write  in  Arabic  characters  the  following : 

1.  Three  hundred  sixteen. 

2.  Two  hundred  forty-seven. 

3.  Three  hundred  eighty-five. 

4.  Four  hundred  nine. 

5.  Two  hundred  thirty-five. 

6.  Six  hundred  eighteen. 

7.  Eight  hundred  forty-two. 

8.  Seven  hundred  fifteen. 

9.  Two  .hundred  forty-three. 

10.  Eight  hundred  nineteen. 

11.  Two  hundred  seven. 

12.  Five  hundred  thirty-nine. 

13.  Four  hundred. 

14.  Six  hundred  nineteen. 

15.  Eight  hundred  twenty. 

16.  Four  hundred  fifty-seven. 

17.  Two  hundred  sixty-nine. 

18.  Three  hundred  eighty-four. 

19.  Five  hundred  sixty-nine. 

20.  Four  hundred  eighty-six. 

21.  Four  tens,  three  units. 

22.  One  hundred,  no  tens,  four  units. 

23.  Two  hundreds,  four  tens,  six  units. 

24.  Five  hundreds,  six  tens,  eight  units. 

PERIODS   AND    ORDERS. 

The  hundreds,  tens,  and  units  composing  a  number 
constitute  what  is  called  a  Period,  and  the  three  places 
are  called  Orders. 

4 


50  ELEMENTARY  ARITHMETIC 

The  figure  in  units'  place  denotes  units  of  the  first 
order  ;  the  figure  in  tens'  place,  units  of  the  second  order  ; 
the  figure  in  hundreds'  place,  units  of  the  third  order. 

EXERCISES. 

1.  Compose  numbers,  using  the  following  units  and 
orders : 

1.  One  unit  of  the  first  order,  two  units  of  the 

second  order,  three  of  the  third. 

2.  Four   units   of  the   first   order,   five   of  the 

second,  six  of  the  third. 

3.  Seven  of  the  first  order,  eight  of  the  second, 

nine  of  the  third. 

4.  None  of  the  first,  two  of  the  second,  three  of 

the  third. 

5.  Four  of  the  second,  five  of  the  first,  six  of 

the  third. 

6.  Seven  of  the  third,  eight  of  the  second,  nine 

of  the  first. 

Suggestion  :  When  an  order  is  not  mentioned  a  cipher  must  be  written 
in  its  place. 

7.  One  unit  of  the  first  order,  two  units  of  the 

third  order. 

8.  Eight  units  of  the  third  order,  one  of  the 

first. 

9.  Five  of  the  third  and  three  of  the  second. 
10.  Nine  units  of  the  third  order. 

2.  Having  written  the  above  numbers,  read  each  one 
of  them,  omitting  and. 

We  have  now  gone  far  enough  to  make  obvious  the 
following 


NOTATION  AND  NUMERATION  51 


PRINCIPLES. 

1.  Ten  units  of  any  order  make  one  unit  of  the 
next  higher  order. 

2.  The  removal  of  a  figure  one  place  to  the  left 
multiplies  its  value  by  ten. 

3.  The  removal  of  a  figure  one  place  to  the  right 
divides  its  value  by  ten. 

4.  O  is  used  to  give  place  and  value  to  the  sig- 
nificant figures. 


NOTE. — Let  the  pupil  use  a  digit  to  illustrate  the  above  principles. 

3.  The  number  next  above  999  is  called  one' thousand 
(1000),  the  1  being  a  unit  of  the  fourth  order. 

The  fourth,  fifth,  and  sixth  orders  complete  a  second 
period,  called  the  period  of  thousands. 

The  seventh,  eighth,  and  ninth  orders  form  the  period 
of  millions. 

The  tenth,  eleventh,  and  twelfth  orders  form  the  period 
of  billions.  Following  billions  are  the  periods  of  trillions, 
quadrillions,  quintillions,  etc. 

4.  Each  period  has  units,  tens,  and  hundreds  of  its 
own. 

5.  Since  ten  units  make  one  ten,  and  ten  tens  make 
one  hundred,  and  ten  hundreds  make  one  thousand,  and 
so  on,  the  Arabic  system  of  notation  is  called  the  decimal 
system.     (Latin,  decem,  ten.) 

6.  The  decimal  system  of  notation  is  best  set  forth  by 
means  of  a  table. 


52  ELEMENTARY  ARITHMETIC 

Table. 

PERIODS.       6th.  5th.  4th.  3d.  2d.  1st. 


>•    j    11 

3    — 

I  o" 


s          a 

NAMES.    -     -     §  ;§  £ 

g  I  1 


§        § 

o  S 


NAME  OF  H,  ^b 

ORDERS.  »  «g  §  g  %  3  <g  j|  ^5 

.WHO*  WHH  WHW  WH§     WHH     WEntS 

NUMBER.        56,  920,  741,  658,     324,     506 


This  number,  as  analyzed  by  the  table,  is  to  be  read 
thus :  Fifty-six  quadrillion,  nine  hundred  twenty  trillion, 
seven  hundred  forty-one  billion,  six  hundred  ffty-eight  million, 
three  hundred  twenty -four  thousand,  five  hundred  six. 

7.  Notice  the  following  features  of  the  number  in  the 
table : 

1.  The   periods   of  the   number   are   set   off  by 

commas. 

2.  All  the  periods  have  three  digits  except  the 

sixth. 

3.  In  reading,  the  name  of  the  first  period  (units) 

is  omitted. 

4.  The  absence  of  significant  figures  is  indicated 

byO. 

NOTE. — A  whole  period  may  be  indicated  by  three  ciphers. 


NOTATION  AND  NUMEEATION 


53 


8.  Name  the  following  periods  : 

1.  First.        Sixth.      Third.       Sixth.       First. 

2.  Second.     First.      Fourth.    Fifth.        Second. 

3.  Third.       Fifth.      Second.    Fourth.     Third. 

4.  Fourth.     Second.  Fifth.        Third.      Fourth. 

9.  Give  the  number  of  the  following  periods : 

1.  Units.  Quadrillions.     Billions. 

2.  Thousands.        Trillions.  Units. 

3.  Millions.  Billions.  Thousands. 

4.  Billions.  Millions.  Trillions. 

5.  Trillions.  Thousands.       Millions. 
10.  Name  the  following  orders  : 

1.  First.  Second.         Fourth.    Thirteenth. 

Third. 
Sixteenth. 
First. 


2.  Fourth. 

3.  Seventh. 

4.  Tenth. 

5.  Second. 


First.        Sixteenth. 
Second.    First. 
Third.      Fourth. 


Thirteenth,  Fourth.    Seventh. 


6.  Thirteenth.  Tenth. 


Seventh.  Tenth. 


NOTE. — Observe  that  the  first,  fourth,  seventh,  tenth,  thirteenth,  and 
sixteenth  orders  name  the  periods  to  which  they  respectively  belong. 

11.  Give  the  names  of  the  following  periods  and  orders  : 


Periods.        Orders. 

1.  6th  and  16th. 

2.  5th  and  13th. 

3.  4th  and  10th. 

4.  3d   and    7th. 

5.  2d  and    4th. 

6.  1st  and    1st. 

7.  6th  and  17th. 

8.  6th  and  18th. 


Periods.        Orders. 

9.  5th  and  15th. 

10.  4th  and  llth. 

11.  4th  and  12th. 

12.  3d   and    8th. 

13.  3d    and    9th. 

14.  2d   and    5th. 

15.  2d    and    6th. 

16.  1st  and    2d. 


54 


ELEMENTAEY  ARITHMETIC 


12.  Copy  carefully  and  point  off  into  periods  : 


1.  5056. 

2.  5565. 

3.  6065. 

4.  46632. 

5.  64645. 

6.  66646. 

7.  54532. 

8.  654653. 

9.  466214. 
10.  664536. 

81 


11.  7385062. 

12.  8237185. 

13.  3785024. 

14.  6083219. 

15.  7882709. 

16.  38420058. 

17.  33468204. 

18.  5284325684. 

19.  7932468412. 

20.  83749275867. 


21.  46825. 

22.  25665. 

23.  54646. 

24.  356236. 

25.  2653665. 

26.  71234567. 

27.  589012345. 

28.  5678010112. 

29.  61314151617. 

30.  618192021222. 


655256234625499844. 


Analyze  and  read  the  first  number,  5056. 

Process.  Analysis. 

5  056  Pointing  off,  we  find  there  are  two  periods :  thousands 

and  units ; — in  the  higher  period,  5  thousands ;  in  the  other, 
0  hundreds,  5  tens,  6  units.  Hence,  the  number  is  read  Five  thousand 
fifty-six. 

Analyze  and  read  the  eleventh  number,  7385062. 

Process.  Analysis. 

7,385,062  Pointing  off,  we  find  there  are  three  periods :    mil- 

lions, thousands,  units.  Beginning  with  the  highest 
period  and  reading,  we  have  Seven  million  three  hundred  eighty-five  thou- 
sand sixty -two. 

In  like  manner  analyze  and  read  each  of  the  numbers 
in  the  first  column. 
Brief  directions  are : 

1.  Copy  the  number. 

2.  Beginning  at  the  right,  point  off. 

3.  Beginning  at  the  left,  read. 

4.  Omit  the  last  name,  units. 


NOTATION  AND  NUMERATION 
13.  Apply  the  directions  to  the  following : 


55 


1.  2725. 

2.  7637. 

3.  2754. 

4.  7237. 

5.  6675. 

6.  4667. 

7.  8263. 

8.  8878. 

9.  7392. 

10.  7940. 

11.  75324. 

12.  61764. 

13.  75773. 

14.  57267. 


15.  82638. 

16.  57188. 

17.  88765. 

18.  38577. 

19.  82683. 

20.  87683. 

21.  7385062. 

22.  8237185. 

23.  3785024. 

24.  6083219. 

25.  7882709. 

26.  8563988. 

27.  6473978. 

28.  5656300. 


29.  7904804. 

30.  9265418. 

31.  575306. 

32.  820583. 

33.  5457474. 

34.  50767576. 

35.  756272376. 

36.  3838785838. 

37.  46887758381. 

38.  500688000233. 

39.  8638866804000. 

40.  97000001543210. 

41.  55555555555555, 

42.  98765432101234. 


14.  Write  in  Arabic  numerals  thirty-six  million  twenty- 
nine  thousand  fifty. 


Process. 


3          2 

m.     th. 


Explanation. 

Since  there  are  three  periods, — millions,  thousands, 
units, — we  write  three  dots  for  each  to  indicate  the 
number  of  places  to  be  filled.  "We  place  36  in  the 
third  period,  29  in  the  second,  and  60  in  the  first. 

The    places    remaining    unoccupied    we    fill    with    ciphers,    and    have 

36,029,050. 


36,029,050 


Brief  directions  are : 


1.  Ascertain  the  number  of  periods  and  orders. 

2.  Beginning  at  the  left,  write  the  significant  figures  of 
each  period,  filling  vacant  places  with  ciphers. 


56  ELEMENTARY  ARITHMETIC 

15.  Apply  the  rule  to  the  following: 

1.  One  thousand  two  hundred  thirty- four. 

2.  Five  thousand  six  hundred  seventy-eight. 

3.  Nine  thousand  twelve. 

4.  Three  thousand  four  hundred  fifty-six. 

5.  Seven  thousand  eight  hundred  ninety. 

6.  One  thousand  two  hundred  thirty-four. 

7.  Nine  thousand  one. 

8.  Three  thousand  three  hundred  forty-four. 

9.  Nine  thousand  nine  hundred. 

10.  Five  thousand  five. 

11.  Eight  million  one  thousand  seventy-six. 

12.  Three  million  one  hundred  sixty-two  thousand  one 
hundred  seventy-two. 

13.  Forty  million  twenty-seven  thousand  six  hundred 
twenty-one. 

14.  One  hundred  twenty-three .  million  four  hundred 
fifty-six  thousand  seven  hundred  eighty-nine. 

15.  Nine   hundred   eighty-seven  million  six  hundred 
fifty-four  thousand  three  hundred  twenty-one. 

16.  Two  hundred  thirty-one  million  two  hundred  two 
thousand  seven  hundred. 

17.  Thirty  million  eight  hundred  twenty-six  thousand 
fifty-one. 

18.  Seven  million  eight  hundred  one  thousand  seventeen. 

19.  Four    hundred    fifty-eight    million    two    hundred 
seventy-five  thousand  six  hundred  ten. 

20.  Ninety  million  five  hundred  seventy-seven  thou- 
sand six  hundred  nineteen. 

21.  Twenty-nine  billion  one  million  one  thousand  eight 
hundred. 


NOTATION  AND  NUMERATION  57 

22.  Fifty-nine  billion  one  million  one. 

23.  Six  hundred  twenty  million  eighty-four  thousand. 

24.  Four  hundred  sixty-seven  million  nine  thousand 
nine. 

25.  Fifty  billion  fifty  million  fifty  thousand  fifty. 

26.  Six  hundred  billion  six  hundred  million  six  hun- 
dred. 

27.  Forty-three    billion    twenty    million    twenty-four 
thousand. 

28.  Eighty-six  billion  eighty-four  thousand  five. 

29.  Five  hundred  twenty-five  million  five. 

30.  Three  hundred  twenty-five  billion  seventeen  mil- 
lion ninety  thousand  nine  hundred  ninety. 

31.  Six    hundred    twenty-five    million    five    hundred 
thousand  twenty-five. 

32.  Seven  hundred  twenty-six  million  eight  thousand 
eight  hundred  eighty. 

33.  Three  hundred  twenty-nine  thousand  three  hun- 
dred three. 

34.  Three  hundred  million  two  hundred  twenty-seven. 

35.  Sixty-six  billion  one  hundred  million  one  hundred 
seventy-four  thousand  seventy-four. 

36.  Eighty  million  eighty  thousand  eight. 

37.  Ninety-five  billion  seven  million  six  thousand  one 
hundred  seventy-five. 

38.  Seventy-six  million  twenty-four. 

39.  Nine  million  twenty-eight  thousand. 

40.  One  million. 

41.  Three  million,  ten. 

42.  Seventy-six  million  twenty-four. 

43.  Four  hundred  million  three  hundred  twenty-nine. 


58  ELEMENTAKY  ARITHMETIC 

44.  Fifty-five  billion  two  hundred  million  nine  hun- 
dred eighty-four  thousand  seventy-five. 

45.  Seventeen  million  four  hundred  seven  thousand 
eighty-four. 

46.  Twenty-eight   million    five    hundred    ninety-four 
thousand  sixty-seven. 

47.  Eighty  million  eighty  thousand  eight. 

48.  Ninety-five  billion  nine  million  eight  thousand  two 
hundred  ninety-four. 

49.  Six  hundred    fifty-five   quadrillion   two    hundred 
fifty-six  trillion  two  hundred  thirty-four  billion  six  hun- 
dred twenty-five  million  four  hundred  ninety-nine  thou- 
sand eight  hundred  forty-four. 

DECIMAL   NOTATION. 

1.  The  Decimal  System  of  notation  extends  to  the 
decimal  parts  of  a  unit,  called  tenths,  hundredths,  thou- 
sandths. 


Decimal  System. 

Ascending  Scale. 

Descending  Scale. 

10 
hundreds, 

10 
tens, 

10 
units, 

Unit 

T*(y  Of            T^J  Of                -fa  Of  a 

a  unit,     a  tenth,      hundredth, 

1000 

100 

10 

1 

.1        .01         .001 

one 
thousand. 

one 
hundred. 

one 
ten. 

0 
N 

E 

one           one              one 
tenth,  hundredth,  thousandth. 

Heading  "by  columns  from  the  unit  towards  the  left,  we  have  the  decimal 
system  of  whole  numhers  ;  reading  by  columns  from  the  unit  towards  the 
right,  we  have  the  decimal  system  of  fractions. 


NOTATION  AND  NUMERATION 


59 


2.  To  mark  the  beginning  of  tenths  and  to  separate 
tenths  and  units  a  sign  [.]  called  the  Decimal  Point  is  used. 

4.235  is  read,  "4  units  and  2  tenths,  3  hundredths, 
5  thousandths";  or,  more  briefly,  "4,  and  235  thou- 
sandths," the  decimal  name  of  the  last  figure  only  being 
repeated. 

Notice  here  that  the  decimal  point  is  read  and.  More- 
over, that  in  reading  numbers  and  is  to  be  applied  to  the 
decimal  point  exclusively. 

3.  When  the  decimal  point  simply  marks  the  begin- 
ning of  a  number  it  is  not  read.     .67  is  read  sixty -seven 
hundredths. 

.5  and  .50  are  of  the  same  value;  but  .05  is  one-tenth 
the  value  of  .5.  [See  Principle  3,  page  11.] 


EXERCISES. 
1.  Read  the  following  decimals : 

11.  .06.        21.  3.38. 

12.  .09. 

13.  .013. 

14.  .075. 

15.  .020. 

16.  .01. 

17.  .001. 

18.  .80. 

19.  .500. 

20.  .625. 


1.  .6. 

2.  .9. 

3.  .13. 

4.  .75. 

5.  .96. 

6.  .548. 

7.  .636. 

8.  .045. 

9.  .008. 
10.  .023. 

2.  "Write  in  figures  : 

1.  Four  tenths.     Eight  tenths.     Nine  tenths. 

2.  Six  hundredths.    Fourteen  hundredths.    Twen- 

ty-one hundredths. 


22.  4.57. 

23.  5.066. 

24.  6.842. 

25.  17.825. 

26.  30.054. 

27.  33.005. 

28.  215.455. 

29.  326.008. 

30.  400.000. 


31.  .303. 

32.  300.007. 

33.  1728.144. 

34.  2150.4. 

35.  3.141. 

36.  365.25. 

37.  .003. 

38.  .300. 

39.  4444.444. 

40.  5000.005. 


60  ELEMENTARY  ARITHMETIC 

3.  Seventeen  thousandths.     Three  hundred  three 

thousandths. 

4.  Eighty-five  hundredths.    Eight,  and  five  tenths. 

5.  Sixty,  and  six  hundredths.     Five,  and  fifteen 

thousandths. 

6.  Nine,  and  two  hundred  sixteen  thousandths. 

7.  Seventy-three,  and  eight  tenths.     One  tenth. 

One  hundredth. 

8.  One   thousandth.      Two   hundredths.      Three 

tenths. 

9.  One  thousand,  and  two  hundredths.      Three 
N     hundred,  and  three  tenths. 

10.  Two  hundred  sixteen,  and  three  hundred  eighty- 
four  thousandths. 

3.  Does  annexing  a  cipher  to  a  decimal  alter  its  value? 

4.  What  is  the  effect  of  inserting  a  cipher  between  the 
decimal  point  and  the  first  figure  of  the  decimal  ? 

5.  What  is  the  effect  of  moving  decimal  figures  towards 
the  right  ? 

UNITED    STATES   MONEY. 

1.  The  decimal   system  of  notation  and   numeration 
applies  to  United  States  currency,  in  which 

10  mills     make  1  cent. 
10  cents     make  1  dime. 
10  dimes   make  1  dollar. 
10  dollars  make  1  eagle. 

2.  Dollars  are  denoted  by  the  sign  $  written  before  the 
number.     $12  means  twelve  dollars. 

3.  The  decimal  point  is  placed  between  dollars  and 
dimes ;  hence,  dimes  are  tenths  of  a  dollar,  cents  are  hun- 
dredths of  a  dollar,  and  mills  thousandths  of  a  dollar. 


NOTATION  AND  NUMERATION  61 

In  practice  the  word  dimes  is  not  much  used;  $1.25  is 
commonly  read,  One  dollar  and  twenty-five  cents.  It 
may  also  be  read,  One  dollar  and  twenty-five  hundredths. 

EXERCISES. 

1.  Read  5.875   and   give   each   figure   its   appropriate 
name. 

2.  Read  $5.875  and  give  each  figure  its  appropriate 
name. 

3.  Read  $5.875  as  dollars,  cents,  and  mills. 

4.  Read  the  following : 

1.  $     .15.  10.  $  16.540.  19.  $329.357. 

2.  $  2.25.  11.  $       .57.  20.  $295.403. 

3.  $  5.375.  12.  $       .625.  21.  $  20.21. 

4.  $  1.060.  13.  $       .375.  22.  $     1.011. 

5.  $18.100.  14.  $     3.94.  23.  $       .10. 

6.  $35.27.  15.  $     3.56.  24.  $       .05. 

7.  $46.39.  16.  $429.384.  25.  $       .01. 

8.  $  3.368.  17.^325.495.  26.  $     4.875. 

9.  $  2.075.  18.  $426.384.  27.  $  16.075. 

5.  Write  the  following : 

1.  Seven  dollars,  forty-nine  cents. 

2.  Fourteen  dollars,  nine  cents. 

3.  Seventy  dollars,  twenty-five  cents. 

4.  Five  dollars,  forty-one  cents,  five  mills. 

5.  Eighty-seven  cents,  five  mills. 

6.  Ninety  dollars,  seven  cents. 

7.  Twenty-seven  dollars,  fifty-six  cents. 

8.  Thirty-seven  dollars,  five  dimes. 

9.  Nine  dollars,  thirty-three  cents,  three  mills. 
10.  Twenty-seven  cents,  eight  mills. 


62  ELEMENTAEY  ARITHMETIC 

11.  Eighty-six  dollars,  five  cents,  two  mills. 

12.  Twenty-one  dollars,  twenty-one  cents,  one  mill. 

13.  Thirty-seven  dollars,  eighteen  cents,  eight  mills. 

14.  Forty-nine  dollars,  nine  cents,  six  mills. 

15.  One  hundred  five  dollars,  seventy  cents. 

16.  Twenty-seven  dollars,  and  four  tenths. 

17.  Three  thousand  dollars,  and  fifty  hundredths. 

18.  Ten  dollars,  ten  cents,  two  mills. 

19.  Twenty-five  dollars,  three  dimes,  seven  cents, 

five  mills. 

20.  One  dollar,  one  cent,  one  mill. 

21.  Twenty  dollars,  twenty  cents,  ten  mills. 

ROMAN    NOTATION. 

The  Roman  System  of  notation  expresses  numbers  by 
means  of  seven  capital  letters,  viz.  : 

Letters:  L,  V.,  X.,  L.,  C.,    D.,     M. 
Values :   1,    5,    10,  50,  100,  500,  1000. 

To  express  other  numbers  these  letters  are  combined : 


Table 

I 

1 

XIV 

.  14 

LX.    . 

...        60 

II 

2 

XV 

15 

LXX 

.    .        70 

III  

...    3 

XVI.  .   .   . 

.    .    .  16 

LXXX. 

...        80 

IV  

...    4 

XVII.     .   . 

.    .    .  17 

XC.    .   . 

...        90 

V  

...    5 

XVIII.  .   . 

.    .    .  18 

C.   .    .   . 

...      100 

VI  

...    6 

XIX.  .   .   . 

.    .    .  19 

CO.    .   . 

...      200 

VII.    .   .   . 

."  .   .    7 

XX.     .   .    . 

.    .    .20 

CCL.     . 

...      250 

VIII.  .    .   . 

...    8 

XXI.   .   .   . 

.    .    .  21 

CCCC.  . 

...      400 

IX  

...    9 

XXIX.    .   . 

.    .    .  29 

D.  .    .   . 

...      500 

X  

...  10 

XXX  

.    .    .  30 

DCC.     . 

...      700 

XI  

.    .    .11 

XXXIV.    . 

.    .    .  34 

M.  .   .   . 

.   .    .    1000 

XII.    .   .   . 

...  12 

XL  

,    .    .  40 

MMM.  . 

.   .   .    3000 

XIII.  . 

.  13 

L. 

.  50 

XVI. 

.  16000 

NOTATION  AND  NUMERATION  63 

The  table  shows  that  combinations  are  made : 

1.  By  repeating  any  of  the  letters,  except  V.,  D.,  and 

L.,  as  IL,  XX. 

2.  By  writing  a  letter  of  less  value  after  one  of  greater 

value,  as  VI,  XV. 

3.  By  writing  a  letter  of  less  value,  except  Y.  and  D., 

before  one  of  greater  value,  as  IX.,  XL. 

4.  By  writing  a  letter  of  less  value  between  two  of 

greater  value,  as  XIV. 

The  letter  standing  before  an  inserted  letter  cannot  be  less  in  value  than 
the  letter  following  it :   XIV.,  not  YIX. 

5.  By  placing  a  bar  over  a  letter  or  a  combination 

of  letters,  as  V  or  XL 

The  effect  of  these  combinations  may  be  stated  briefly 
as  follows : 


PRINCIPLES. 

1.  Repeating  a  letter  repeats  value,  as  in  XX. 

2.  Affixing  a  letter  increases  value,  as  in  XI. 

3.  Prefixing  a  letter  diminishes  value,  as  in  IX. 

4.  Inserting  a  letter  diminishes  value,  as  in  XIX. 

5.  Placing  a  bar  increases  value,  as  in  VI. 


EXERCISES. 
1.  Let  the  principles  answer  the  following  questions : 

1.  What    does    the    combination    III.    express? 

"Why? 

2.  What    does    the    combination    IV.    express? 

Why? 


64 


ELEMENTARY  ARITHMETIC 


3.  What    does    the    combination    XV.    express? 

Why? 

4.  What   does  the   combination    XIV.   express? 

Why? 

5.  What  does  VIX.  express  ?     Why  ? 

6.  How  is  14  properly  expressed  ? 

7.  What  is  the  value  of  MIX.  ?     Why  ? 

8.  What  is  the  value  of  LIX.  ?     Why  ? 

9.  What  is  the  value  of  IXL.  ?     Why  ? 

10.  What  is  the  value  of  LXL  ?     Why? 

11.  Is  it  a  letter  of  less  value  or  of  greater  value 

that  is  prefixed,  affixed,  or  inserted  ? 

12.  Can  a  letter  before  an  inserted  letter  be  less 

than  the  one  following  it  ? 
2.  Read  the  following : 


1.  XX. 

2.  XXI. 

3.  XXIV. 

4.  XXV. 

5.  XXVI. 

6.  XXVHI. 

7.  XIX. 

8.  XXXII. 

9.  XLV. 

10.  LXV. 

11.  XXXV. 

12.  XL. 

13.  LX. 

14.  LXX. 

15.  LVIII. 

16.  LXXX. 


17.  LXIX. 

18.  IXIV. 

19.  XC. 

20.  XCIV. 

21.  XCIX. 

22.  XLIV. 

23.  CDXX. 

24.  CCXXIV. 

25.  VIII. 

26.  XXIX. 

27.  MDLIV. 

28.  XXXIII. 

29.  CIX. 

30.  CXI. 

31.  CXLV. 

32.  CCXIX. 


33.  XCI. 

34.  DCXC. 

35.  DCCX. 

36.  XXXIX. 

37.  LXXXIX. 

38.  CLXXIX. 

39.  CCCCXIVH. 

40.  VIII. 

41.  CCXC. 

42.  CXLIX. 

43.  CLI. 

44.  MD. 

45.  CIX. 

46.  LXV. 

47.  E 

48.  xiv: 


ADDITION  65 

3.  Express  in  Roman  notation  : 

1.  25.   11.  58.   21.  387.  31.  25.  41.  100. 

2.  45.   12.  97.   22.  587.  32.  46.  42.  2000. 

3.  16.   13.  324.   23.  436.  33.  52.  43.  2505. 

4.  38.   14.  423.   24.  789.  34.  87.  44.  3333. 

5.  72.   15.  520.   25.  207.  35.  77.  45.  4444. 

6.  47. v  16.  337.   26.  999.  36.  66.  46.  5555. 

7.  95.   17.  495.   27.  1000.  37.  56.  47.  6666. 

8.  69.   18.  327.   28.  1500.  38.  51.  48.  7777. 

9.  73.   19.  514.   29.  1550.  39.  49.  49.  8888. 
10.  46.   20.  599.   30.  1555.  40.  99.  50.  1892. 


ADDITION. 

Oral. 

INDUCTIVE   STEPS. 
How  many  are : 

1.  2  cents  and  2  cents  ?     2  and  2  ? 

2.  3  pencils  and  2  pencils  ?     3  and  2  ? 

3.  4  doors  and  3  doors  ?     4  and  3  ? 

4.  5  windows  and  3  windows  ?     5  and  3  ? 

5.  6  birds  and  4  birds  ?     6  and  4  ? 

6.  4  birds  and  6  birds  ?     4  and  6  ? 

7.  5  fingers  and  5  fingers  ?     5  and  5  ? 

8.  7  fields  and  3  fields  ?     7  and  3  ? 

9.  7  fields  and  3  trees  f 

Since  7  and  3  are  ten,  why  cannot  you  say  that  7  fields 
and  3  trees  are  10  ?     What  is  the  unit  of  each  number  ? 

Then  we  must  conclude  that  only  numbers  having  the 
same  unit  can  be  expressed  in  a  single  number. 

5 


66  ELEMENTARY  ARITHMETIC 

DEFINITIONS. 

1.  Numbers  having  units  of  the  same  kind  are  called 
Like  Numbers. 

2.  Addition  is  the   process  of  uniting  two  or  more 
numbers  into  a  single  number,  called  their  Sum. 

3.  The  sum,  then,  is  the  result  obtained  by  adding. 

4.  The  Sign  of  Addition  is  an  upright  cross  (-J-)  called 
plus  (more);    it  is  placed  between  the  numbers  to  be 
added.     7  -f  4  is  read  "  7  plus  4." 

5.  The  Sign   of   Equality  is  two  parallel  horizontal 
lines,  =.     7  +  4  =  11  is  read  "  7  plus  4  equals  11." 

6.  7  +  4  =  11,  being  an  expression  of  equality,  is  called 
an  Equation. 


PRINCIPLES. 

1.  Only  like  numbers  can  be  added. 

2.  The  sum   and   the   numbers    added   are   like 
numbers. 


EXERCISES. 

1.  The  digits  are:   1,  2,  3,  4,  5,  6,  7,  8,  9,  0.     What 
numbers  do  they  represent  ? 

2.  Answer  the  following  questions  at  sight : 


1.  0 

+ 

1 

=  ? 

8. 

7  + 

0  = 

? 

15. 

4 

+  5 

=  ? 

2.  1 

+ 

3 

9 

9. 

8  + 

1  = 

? 

16. 

5 

+  6 

9 

3.  2 

+ 

2 

_  9 

10. 

9  + 

«)  

? 

17. 

6 

+  4 

—  ? 

4.  3 

+ 

1 

9 

11. 

0  + 

4  = 

? 

18. 

7 

+  5 

—  ? 

5.  4 

+ 

3 

9 

12. 

1  + 

5  = 

? 

19. 

8 

+  6 

=  ? 

6.  5 

+ 

1 

__  9 

13. 

2  + 

6  = 

? 

20. 

9 

+  4 

=  ? 

7.  6 

+ 

2 

_  9 

14. 

3  + 

4'= 

? 

21. 

0 

+  7 

9 

ADDITION  67 


22.  1  +  8  =  ?' 

29.  8  +  9  =  ?        36.  5  +  5  =  ? 

23.  2  +  9  =  ? 

30.  9  -f-  7  =  ?        37.  6  +  6  =  ? 

24.  3  -f  7  =  ? 

31.  0  +  0  =  ?        38.  7  +  7  =  ? 

25.  4  +  8  =  ? 

32.  1  +  1  =  ?        39.  8  +  8  =  ? 

26.  5  +  9  =  ? 

33.  2  +  2  ==  ?        40.  9  +  9  =  ? 

27.  6  +  7  =  ? 

34.  3  -f  3  =  ?        41.  9  +  2  =  ? 

28.  7  -f  8  '==  ? 

35.  4  -f  4  —  ?        42.  9  +  6  =  ? 

(A.) 

(B.)                            (C.) 

1.  3,  2,  5.         6. 

3,  4,  2,  5.       11.  5,  4,  3,  7. 

2.  4,  3,  2.         7. 

2,  3,  4,  6.       12.  6,  8,  4,  3. 

3.  5,  4,  3.         8. 

6,  3,  2,  4.       13.  2,  9,  6,  2. 

4.  6,  2,  4.         9. 

5,  7,  3,  6.       14.  4,  6,  8,  3. 

5.  5,  3,  2.       10. 

8,  4,  2,  3.       15.  7,  8,  4,  9. 

3.  Begin  at  the  uppermost  3  in  A,  and  give  orally  and 
at  sight  the  sum  of  the  numbers  in  each  line  thus :  3,  5, 
10,  sum  of  first  line  of  A. 

4.  Begin  at  the  same  place  and  give  in  the  same  man- 
ner the  sum  of  each  column  thus:  3,  7,  12,  18,  83,  the 
sum  of  the  first  column  of  A. 

ORAL   PROBLEMS. 

1.  How  many  fingers  are  6  fingers  and  4  fingers? 

Answer:  6  fingers  -(-  4  fingers  =  10  fingers. 

2.  If  an  apple  cost  3  cents  and  an  orange  5  cents,  how 
much  did  both  cost  ? 

3.  A  girl  paid  6   cents  for  a  pencil  and  5  cents  for 
sugar-plums.     How  much  did  she  pay  for  both  ? 

4.  A  boy  spent  7  cents  for  some  paper  and  5  cents  for 
an  orange.     How  much  did  he  spend  for  both  ? 

5.  A  man  rode  on  his  bicycle  8  miles  the  first  hour 


68  ELEMENTARY  ARITHMETIC 

and  7  miles  the  second  hour.    How  far  did  he  ride  in  the 
two  hours  ? 

6.  A  horse  travels  9  miles  the  first  hour  and  8  miles 
the  second  hour.     How  far  does  he  go  in  two  hours  ? 

7.  A  farmer  who  had  5  horses,  bought  5  more  at  one 
time,  and  at  another  time  4  more.     How  many  did  he 
then  have  ? 

8.  A  matron  bought  8  quarts  of  strawberries  of  one 
man,  5  quarts  of  another,  and  5  of  another.     How  many 
quarts  did  she  buy  in  all  ? 

9.  A  traveller  bought  for  lunch   5  cents'  worth  of 
crackers,  4  cents'  worth  of  milk,  and  8  cents'  worth  of 
berries.     What  did  his  lunch  cost  him  ? 

10.  How  many  animals  are  in  a  field,  there  being  6 
cows,  4  oxen,  and  8  sheep  ? 

11.  Margaret  gave  $2.00  for  an  atlas,  $3.00  for  a  his- 
tory, and  $8.00  for  a  dictionary.     How  much  did  she  pay 
for  all? 

12.  A  man  earns  $9.00,  while  a  boy  and  a  girl  earn 
each  $5.00.     How  much  do  all  earn  when  the  man  has 
earned  $9.00  ? 

13.  Add  2,  6,  8,  7,  1,  2.     Also,  8,  5,  3,  2,  4,  9. 

14.  A  house  has  8  windows  on  the  east  side,  7  on  the 
west,  and  9  on  the  south.     How  many  are  there  in  all  ? 

15.  A  rose-bush  has  on  one  branch  8  roses,  on  another 
7  roses,  and  on  a  third  6  roses.     How  many  roses  on  the 
three  branches  ? 

16.  In  one  part  of  a  building  there  are  9  offices,  in 
another  part  5  offices,  in  a  third  part  6  offices.     How 
many  in  the  three  parts  ? 

17.  Jennie  sold  to  a  lady  4  quarts  of  berries,  8  quarts 


ADDITION  69 

of  peaches,  2  quarts  of  beans,  and  1  quart  of  peas.     How 
many  quarts  did  the  lady  buy  ? 

18.  A  gentleman  selected  for  a  bouquet  8  white  carna- 
tions, 6  pink  ones,  and  6  red  ones.     How  many  in  all  in 
the  bouquet  ? 

19.  Sarah  wrote  on  Monday  9  letters,  on  Tuesday  8 
letters,  on  Wednesday  7  letters.     How  many  letters  did 
she  write  in  the  three  days  ? 

20.  A  foot-ball  team  won  2-  games  in  September,  3 
games  in  October,  4  games  in  November,  not  counting 
the  game  won  on  Thanksgiving  Day.     How  many  games 
were  won  in  the  three  months  ? 

21.  Count  by  2's  from  0  to  50;  thus:  0,  2,  4,  6,  etc. 

22.  Count  by  3's  from  0  to  51.     From  4  to  47. 

23.  Count  by  4's  from  0  to  52.     From  5  to  59. 

24.  Count  by  6's  from  0  to  72.     From  5  to  77. 

25.  Count  by  7's  from  0  to  84.     From  6  to  90. 

26.  Count  by  8's  from  0  to  96.     From  4  to  100. 
37.  Count  by  9's  from  7  to  115.     From  0  to  117. 

Addition  of  Single  Columns. 

WRITTEN  EXERCISES. 
1.  What  is  the  sum  of  3,  6,  8,  7  ? 

Process.  Explanation. 

3  Explain,  speaking  thus:  "For  convenience,  we  write  the 

g  numbers  in  a  column.     "We  then  add  thus  :  7  -(-  8  =  15 ;  15 

-f  6=  21 ;  21  +  3  =  24,  the  sum." 

To  prove  the  correctness  of  the  result,  we  add  the  column 
7  downwards :    3  +  6  =  9;    9  +  8=17;    17  +  7  =  24,  the 

24  sum.     We  may  explain  more  briefly,  saying  «(  3,  9,  17,  ££; 

again,  7,  15,  21,  &#.» 


70  ELEMENTARY  ARITHMETIC 

What  is  the  first  principle  of  Addition  f 

Are  the  preceding  added  numbers  like  numbers  f 

They  are  alt  units  of  what  order  ? 

2.  In  a  similar  manner  add  and  explain  these : 

(1.)        (2.)       (3.)        (4.)       (5.)       (6.)       (7.)        (8.)       (9.)       (10.) 

7594874539 
5737928978 
8683  £53182 

4    4    4    5   A   JL  A   A   JL  A 

(11.)  (12.)  (13.)  (14.)  (16.)  (16.)  (17.)  (18.)  (19.)  (20.) 
6866836522 
3533754979 
5591424346 
9    7    9    5    4    7    86    44 


(21.)   (22.)  (23.)   (24.)   (25.)  (26.)  (27.)   (28.)   (29)  (30.) 


2 

7 

3 

5 

9 

6 

5 

9 

4 

2 

9 

8 

8 

2 

7 

6 

8 

7 

8 

9 

6 

2 

6 

6 

3 

7 

7 

6 

6 

8 

4 

6 

1 

8 

1 

2 

6 

8 

2 

4 

7- 

5 

9 

3 

5 

4 

9 

3 

8 

5 

(31.) 

(32.) 

(33.) 

(34.) 

(35.) 

(36.) 

(37.) 

(38.) 

(39.) 

(40.) 

6 

7 

3 

9 

5 

5 

7 

6 

9 

8 

9 

4 

9 

6 

6 

2 

3 

3 

4 

7 

7 

1 

2 

5 

4 

8 

5 

8 

7 

5 

5 

6 

7 

2 

7 

6 

8 

6 

3 

5 

8 

3 

5 

8 

1 

3 

6 

4 

6 

3 

ADDITION  71 

3.  Required  the  sum  of  the  following  numbers  : 

1.  4,5,2,4,8,6.  11.  6,9,7,5,8,4. 

2.  7,  6,  4,  8,  5,  3.  12.  7,  4,  1,  6,  3,  8. 

3.  9,  3,  5,  6,  4,  2.  13.  3,  9,  2,  7,  5,  1. 

4.  8,  6,  8,  5,  2,  3.  14.  9,  6,  5,  2,  8,  7. 

5.  4,  3,  8,  6,  5,  4.  15.  5,  6,  4,  7,  1,  8. 

6.  8,  9,  3,  1,  9,  4.  16.  5,  2,  8,  6,  3,  4. 

7.  6,  7,  9,  4,  5,  6.  17.  7,  3,  5,  8,  6,  1. 

8.  4,  9,  3,  6,  8,  2.  18.  9,  5,  9,  4,  6,  8. 

9.  7,  3,  4,  8,  2,  5.  19.  3,  6,  8,  5,  6,  3. 
10.  4,  7,  5,  6,  2,  4.  20.  7,  4,  9,  4,  9,  3. 

WRITTEN  PROBLEMS. 

1.  A  book-agent  sold  3  books  every  day  for  six  days. 
How  many  books  did  he  sell  in  the  six  days  ? 

34.3  +  8  +  3  +  3  +  3  =  number  sold. 

NOTE. — Let  the  pupil  indicate  the  solution  of  each  problem  in  this 
way. 

2.  In  a  family  are  six  children;    the  youngest  is  2 
years   old.     Each  of  the    other   five   is    2    years    older 
than  the  one  next  younger.     What  is  the  age  of  the 
oldest  ? 

3.  A  tree  was  broken  off  9  feet  from  the  ground; 
8  feet  of  the  piece  broken  off  lies  upon  the  ground,  and 
the  remaining  7  feet  of  it  lies  in  water.     How  high  was 
the  tree? 

4.  A  book-case  contained  5  shelves.     The  librarian 
removed  9  books  from  the  fifth  shelf,  8  books  from  the 
fourth,  5  books  from  the  third,  7  books  from  the  second, 
and  4  books  from  the  first.     How  many  books  did  he 
remove  from  the  case? 


72  ELEMENTARY  ARITHMETIC 

5.  A  man  on  examining  his  purse  found  therein  5 
dimes,  4  quarter-dollars,  3  dollars  (silver),  2  half-dollars, 
and  one  eagle.     How  many  pieces  of  money  had  he  ? 

6.  I  have  three  different  measures :  gill,  pint,  quart. 
The  pint  contains  4  gills,  the  quart  8  gills.     How  many 
gills  do  the  three  measures  contain  ? 

7.  If  a  pint  of  water  weighs  one  pound,  and  2  pints 
equal  1  quart,  and  8  pints  equal  1  gallon,  how  much  does 
the  water  in  a  pint,  a  quart,  and  a  gallon  measure  weigh  ? 

8.  John,  James,    George,   and   Henry   were   solving 
problems.     John  solved  5,  James  solved  4,  George  as 
many  as  both  John  and  James.     Henry  copied  the  4  that 
James  did.     How  many  problems  were  solved  ? 

9.  A  room  is  9  feet  long,  9  feet  wide,  and  8  feet  high. 
How  long  a  line  would  measure  twice  across  the  width 
of  the  room  and  then  up  to  the  ceiling  ? 

10.  Some   articles   on   my  desk   measure  as  follows : 
A  pen-holder,  7  inches;  a  steel  ink-eraser,  7  inches;  a 
pair  of  dividers,  6  inches;    a  lead-pencil,  5  inches;   a 
paper-cutter,  8   inches;    and   a   fountain-pen,  7   inches. 
Find  the  combined  length  of  the  six  articles. 

11.  On  a  wintry  holiday  6  boys  and  4  girls,  set  free 
from  school,  went  skating ;  7  girls  and  5  boys  went  coast- 
ing ;  4  boys  built  a  snow  man,  and  2  a  snow  fort.     Find 
the  number  of  boys,  the  number  of  girls,  and  the  number 
of  pupils  the  school  contained. 

12.  A  street-vender  sold  to  some  boys  4  apples  for 
8  cents,  3  oranges  for  9  cents,  2  pears  for  6  cents,  5  lemons 
for   9    cents,  and   2   peaches   for   4   cents.      How  many 
units  of  fruit  did  he  sell,  and  how  many  cents  did  he 
receive  ? 


ADDITION  73 

13.  A  school -girl  wrote  in  her  copy-book  7  lines  on 
Monday,  8  lines  on  Tuesday,  6  lines  on  "Wednesday,  5 
lines  on  Thursday,  and  7  lines  on  Friday.     How  many 
lines  did  she  write  during  the  week  ? 

14.  Find  the  sum  of  1,  2,  3,  4,  5,  6,  7,  8,  9. 

15.  Change  to  Arabic  numerals  the  following  Roman 
numbers,  and  find  their  sum  :  III.,  IX.,  VIIL,  V.,  IV.,  VI. 

ORAL   EXERCISES. 

1 .  How  many  are  : 

1.  10  cents  and  10  cents?     10  and  10? 

2.  10  dollars  and  20  dollars?     10  and  20? 

3.  10  tops  and  30  tops  ?     10  and  30  ? 

4.  30  books  and  10  books  ?     30  and  10  ? 

5.  35  books  and  10  books  ?     35  and  10  ? 

6.  10  books  and  40  books  ?     10  and  40  ? 

7.  40  men  and  10  men  ?     40  and  10  ? 

8.  40  men  and  8  men  ?     40  and  8  ? 

9.  10  men  and  18  men  ?     10  and  18  ? 
10.  20  ships  and  16  ships  ?     20  and  16  ? 

2.  Count  by : 

1.  10's  from  0  to  100.     From  2  to  92. 

2.  10's  from  5  to  95.     From  4  to  94. 

3.  10's  from  7  to  97.     From  3  to  93. 

4.  10's  from  8  to  98.     From  19  to  90. 

5.  10's  from  9  to  99.     From  1  to  101. 

6.  20's  from  0  to  100.     From  5  to  85. 

7.  20's  from  2  to  102.     From  6  to  86. 

8.  20's  from  3  to  103.     From  7  to  87. 

9.  30's  from  5  to  95.     From  8  to  98". 
10.  30's  from  10  to  100.     From  9  to  99. 


74  ELEMENTARY  ARITHMETIC 

Addition  of  Several  Columns. 

•WRITTEN  EXERCISES. 
1.  Find  the  sum  of  $357,  $470,  $534. 

Process.  Explanation. 

$357  Explain,  speaking  thus:  "  Since  the  numbers  express  the 

Anr\  same  kind  of  thing,  they  are  like  numbers,  and  their  sum 
can  be  found.  In  accordance  with  the  same  principle,  we 
write  units  of  the  same  order  in  the  same  column,  and  pro- 


$1361         ceed  thus :  4  units  ~\-  7  units  =  11  units  =  1  ten  and  1  unit ; 
1  ten  -f-  3  tens  -f-  7  tens  -f-  6  tens  =16  tens  =  1  hundred 
and  6  tens ;    1  hundred  -f-  5  hundred  -f-  4  hundred  -f-  3  hundred  =  13 
hundred  =  1  thousand  and  3  hundreds.     Hence,  the  sum  is  $1361." 

Repeat  the  principle  on  which  your  work  depends. 

What  is  a  unit  ?  What  is  the  unit  of  each  of  the  numbers 
added  ?  What  is  the  unit  of  their  sum  ? 

Can  the  unit  of  the  numbers  added  and  the  unit  of  their  sum 
ever  be  different? 

Brief  directions  are : 

1.  "Write  units  of  the  same  order  in  the  same  column. 

2.  Begin  at  the  right  to  add. 

3.  Add  the  columns  separately. 

4.  When  tens  of  any  order  appear,  carry  them  to  the 
next  column. 

5.  Write  the  entire  sum  of  the  last  column. 

6.  Prove  by  adding  each  column  in  reverse  order. 


2.  Add  and  prove  each  of  these : 

(1.)            (2.)            (3.)           (4.)            (5.)  (6.)  (7.)  (8.) 

435   340   629   375   390  238  4  310 

576  -  457  •  308   420   534  354  44  73 

357   328    594   965   728  96  444  206 


ADDITION  75 


(9.) 
$132 

(10.) 
$12.09 

(ii.) 
$62.22 

(12.) 

$2.42 

(13.) 

$27.16 

(14.) 

$43.38 

108 

13.17 

119.93 

6.54 

36.59 

437.84 

159 

4.60 

7.37 

107.31 

19.31 

394.38 

122 

4.18 

8.04 

3.52 

45.42 

517.73 

81 

2.58 

14.17 

6.09 

23.08 

654.85 

106 

1.61 

13.71 

11.56 

24.25 

826.24 

When  dollars  and  cents  are  written  in  columns,  do  not  the 
decimal  points  form  a  column  ? 

Did  you  place  a  decimal  point  in  each  sum  directly  under 
the  column  of  points  ? 

3.  Add  and  prove  each  of  these : 

(i.)  (2.)  (3.)  (4.) 

Soldiers.  Bushels.  Units.  Miles. 

42790  5279  3078  1 

42730  570  596  96 

4203  4369  9  538 

537  520  1034  38 

48  210  1005  176 

1  22  3333  4444 

4.  Add  4795,  3084,  3970,  6952,  7964. 

5.  Add  5968,  3075,  493,  3980,  77. 

6.  Add  2325,  2642,  5236,  8230,  3616,  21. 

7.  Add  4836,  658,  816,  636,  1158,  6. 

8.  Add  52055,  3650,  62055,  4268,  13670,  231. 

9.  Add  46632,  5056,  64645,  5565,  66646,  54532. 

10.  Add  $75.32,  $27.25,  $76.37,  $27.54,  $72.37. 

11.  Add  $617.64,  $763.55,  $377.07,  $767.76,  $735.67. 

12.  Add  $767.37,  $54.76,  $67.54,  $476.71,  $32.55. 


7 


6  ELEMENTARY  ARITHMETIC 

13.  Add  $67.645,  $75.464,  $57.677,  $74.766,  $67.257. 

14.  Add  $82.638,  $82.63,  $571.88,  $88.765,  $7.87. 

15.  Add  $472.49,  $57.912,  $7.392,  $93.494,  $79.40. 

16.  Add  the  following: 

1.  Fifty-four  thousand,  two  thousand  sixty-five. 

2.  Nine  thousand,  six  thousand  seventy-one. 

3.  Eighteen  thousand,  six  thousand  four. 

4.  Thirty-five  thousand,  four  hundred  thousand. 

5.  Two  million,  three  million,  four  thousand,  867. 

6.  Eight  million,  five  hundred  thirty-nine  million. 

7.  Twenty-one  million,  eight  million,  788. 

8.  Four  hundred,  twenty-nine  thousand,  500. 

9.  Nine  million ,  eight  hundred,  five  thousand,  639. 
10.  Eight  hundred,  twenty-one  thousand,  forty- 
seven. 

17.  Find  the  sum  of  the  following : 

1.  Twenty-seven  dollars,  fifty-six  cents;  ninety 
dollars,  twenty  cents;  seventy-five  dollars, 
twenty-seven  cents;  nine  hundred  twenty- 
seven  dollars,  eighty  cents ;  thirty-nine  dol- 
lars, fifty  cents. 

2.  Three  hundred  dollars,  nine  cents ;  twenty-nine 

dollars,  seven  cents;  five  dollars,  forty-one 
cents,  five  mills ;  forty  dollars,  seventy-nine 
cents;  seven  dollars,  sixty-two  cents,  five 
mills. 

3.  Four   hundred   twenty-nine   dollars,   seventy 

cents;  seventy  dollars,  twenty-five  cents; 
eighty-seven  cents,  five  mills ;  one  hundred 
sixty-nine  dollars,  twenty-eight  cents ;  thirty- 
three  dollars,  sixty-seven  cents,  five  mills. 


ADDITION  77 

4.  Five  hundred  sixty  dollars,  eighty- two  cents ; 

nine  hundred  eighty-seven  dollars;  ninety 
dollars  twenty  cents. 

5.  Six  hundred  dollars,  eighty-nine  cents;  one 

hundred  dollars,  forty-seven  cents;  two  hun- 
dred thirty-four  dollars,  eighty  cents;  five 
hundred  fifty-four  dollars,  ninety  cents ; 
nine  dollars,  thirty-one  cents,  two  mills 
five-tenths  of  a  mill. 

WRITTEN  PROBLEMS. 

1.  A  man  owns   $7580  in  land,  $4750  in  live  stock, 
$1675  in  notes,  and  $2987  in  cash.      How  much  is  he 
worth  ? 

NOTE. — Indicate  all  processes,  using  proper  signs  ;  thus,  $7580  -(-  $4750 
+  $1675  -f  $2987  =  Amount. 

2.  At  the  battle  of  Gettysburg  the  loss  in  the  Union 
army  was  2,834  men  killed  and  13,790  wounded;  and  in 
the  Confederate  army,  4,500  killed  and  26,500  wounded. 
What  was  the  entire  loss  in  both  armies  ? 

3.  A  man  paid  $375  for  a  carriage,  $250  for  a  horse, 
and  $175  for  harness.     How  much  did  he  pay  for  all  ? 

4.  A  grocer  bought  some  sugar  for  8  dollars  and  some 
tea  for  7  dollars.     What  amount  will  he  receive  for  the 
two  that  he  may  gain  6  dollars  ? 

5.  Suppose  a  merchant  has  3756  dollars  in  bank-bills, 
4793  dollars  in  gold,  264  dollars  in  silver,  and  5  dollars 
in  cents.     How  much  money  has  he  ? 

6.  The  distance  from  A.  to  B.  is  370  miles,  from  B. 
to  C.  465  miles,  from  C.  to  D.  329  mrles.     How  far  is  it 
from  A.  to  D.  ? 


78  ELEMENTARY  ARITHMETIC 

7.  I  received  $53.70  for  105  bushels  of  oats  and  $73.20 
for  112  bushels  of  corn.    How  much  money  did  I  receive  ? 
How  many  bushels  did  I  sell  ? 

8.  In  1870  the   population  of  New  York  City  was 
942,292;  of  Philadelphia,  674,022 ;  of  Brooklyn,  396,099; 
of  St.  Louis,  310,864;  and  of  Chicago,  298,977.     What 
was  the  total  population  of  those  cities  ? 

9.  I  paid  away  $396  in  gold,  $574  in  silver,  and  $413 
in  notes,  and  had  $413  left.     How  many  dollars  had  I  at 
first? 

10.  A  man   making  his  will  left  $3450  to  his  wife, 
$2675  to  his  oldest  son,  $1850  to  his  second  son,  and 
$1290  to  his  youngest  son.     What  amount  of  money  was 
bequeathed  in  his  will  ? 

11.  Abraham  lived  175  years;  Isaac,  180;  Jacob,  147; 
Joseph,  110;   Moses,  120;  Joshua,  110.     Find  the  sum 
of  their  ages. 

12.  A  coal-dealer  furnished  me  8  loads  of  coal,  weigh- 
ing as  follows:  2120  pounds,  2312  pounds,  2218  pounds, 
1927  pounds,  2063  pounds,  2284  pounds,  1995  pounds, 
1987  pounds.     What  was  the  weight  of  all  ? 

13.  B.  paid  $650  for  a  lot,  on  which  he  built  a  house 
for  $5785.     The  fence  around  the  lot  cost  $167,  the  pave- 
ment cost  $243,  and  the  plumbing  cost  $315.    Selling  the 
property,  he  gained  $1200.     How  much  did  he  receive 
for  it? 

14.  I  bought  40  pounds  of  sugar  for  187  cents,  10 
pounds  of  coffee  for  273  cents,  and  75  pounds  of  rice  for 
357  cents.     How  many  cents  did  I  give  for  all  ?     How 
many   dollars   and   cents?      How   many   pounds   did  I 
buy? 


ADDITION  79 

15.  Add: 


(1.) 

(2-) 

(3.) 

(4-) 

(5.) 

923 

436 

9945 

5967 

LXXX. 

548 

597 

6878 

4304 

XC. 

796 

435 

5904 

5706 

C. 

837 

608 

9237 

9708 

CO. 

537 

957 

9705 

9543 

CCL. 

943 

890 

6342 

5556 

D. 

324 

457 

4356 

4395 

XIV. 

534 

565 

5274 

4950 

XVI. 

798 

402 

9252 

3076 

XIX. 

932 

917 

6244 

2705 

XXIX. 

456 

393 

4270 

8394 

XXXIV. 

539 

407 

9396 

7657 

XLVH. 

798 

039 

6250 

3203 

LXXXV. 

630 

575 

4396 

4308 

MMM. 

235 

970 

5394 

9867 

DLXXX. 

497 

875 

4379 

9549 

LXXVI. 

538  ' 

579 

6830 

6870 

DCCC. 

925 

689 

5364 

4396 

XM. 

989 

772 

8458 

9244 

MXC. 

922 

685 

7793 

8859 

MDCCCXCVm. 

REVIEW. 

1.  Repeat  the  principles  of  the  Roman  system  of  nota- 
tion. 

2.  Name  the  periods  given  in  the  Arabic  numeration 
table. 


80  ELEMENTARY  ARITHMETIC 

SUBTRACTION. 

Oral. 
INDUCTIVE   STEPS. 

1.  How  many  are : 

1.  4  apples  less  2  apples? 

2.  5  dollars  less  3  dollars  ? 

3.  6  birds  less  3  birds  ? 

4.  7  cents  less  4  cents  ? 

5.  10  dollars  less  5  dollars  ? 

2.  What  is  the  difference  : 

1.  Between  9  years  and  6  years? 

2.  Between  10  days  and  8  days  ? 

3.  Between  11  dollars  and  7  dollars? 

4.  Between  11  dollars  and  7  yards? 

How  must  the  question  be  changed  that  the  difference 
may  be  found  ? 

Between  what  kind  of  numbers  can  we  find  a  differ- 
ence? 

3.  The  difference  between  11   and  7  is  four.     If  you 
add  the  difference  4  to  the  less  number,  what  number  do 
you  obtain? 

4.  What  is  the  difference  between  12  days  and  7  days? 
How  can  you  prove  that  5  days  is  the  correct  result  ? 
Is  not  finding  the  difference  between  12  and  7  the  same 

as  subtracting  7  from  12? 

DEFINITIONS. 

1.  Subtraction  is  the  process  of  finding  the  difference 
between  two  numbers.  In  subtraction  one  number  is 
said  to  be  taken  from  another. 


SUBTRACTION  81 

2.  The  Minuend  is  the  number  from  which  another 
number  is  to  be  subtracted. 

3.  The  Subtrahend  is  the  number  to  be  subtracted. 

4.  The  Remainder  or  Difference  is  the  result  obtained. 

5.  The  Sign  of  Subtraction  is  a  short  horizontal  line 
( — )  called  minus  (less);    it  is  placed  between  the  two 
numbers  to  be  subtracted,  and  shows  that  the  one  after 
it  is  to  be  subtracted  from  the  one  before  it. 

11  —  7  =  4,  is  read,  "11  minus  7  equals  4." 
11  —  7  =  4,  being  an  expression  of  equality,  is  called 
what? 


PRINCIPLES. 

1.  The  minuend  and   subtrahend  must  be  like 
numbers. 

2.  The   sum  of  the   subtrahend   and  remainder 
equals  the  minuend. 


DRILL. 
Complete,  at  sight,  the  following  equations  : 

1.  4  —  1  =       5  —  2  =       7  —  2  =       10  —  7  = 

2.  4  —  3  =  •    5  —  4=       9  —  6=         9  —  4  = 

3.  4  —  2  =       5  —  3=       9  —  3=         8  —  5  = 

4.  4 --4=       5  —  5=       9  —  9=         8  —  8  = 

From       10        9        6      11      12      14      16      15      10        9 
Subtract_6     _7_310_5J7_6_8jt_7 

From       20      18      13      13      14      15      11      10        8      13 
Subtract  10989679875 


82  ELEMENTARY  ARITHMETIC 

From       12  19      18      19      15      13      16  18      20     14 

Subtract_6  _H)_9_9874626 

From       15  16      13      11      12      12      13  11      15      15 

Subtract    5  __8        5        7        8        7        6  8        612 

From       16  16      17      17      18      18      19  19      20     20 

Subtract    69855      10      15  739 


ORAL   PROBLEMS. 

1.  A  boy  solved  12  examples  on  Monday  and  15  on 
Tuesday.    How  many  more  did  he  solve  on  Tuesday  than 
on  Monday  ? 

2.  Mr.  A.  had  20  sheep,  but  sold  10.     How  many  were 
left? 

3.  Henry  had   50   apples;    he   gave   30   of  them   to 
Charles.     How  many  had  he  left  ? 

4.  A  merchant  had  19  metres  of  cloth  and  sold  8 
metres.     How  many  metres  had  he  left  ? 

5.  John  had  28  cents,  but  gave  8  to  James.     How 
many  remained  ? 

6.  Jenkins  is  80  years  old  and  his  son  is  50.     How 
much  older  is  Jenkins  than  his  son  ? 

7.  Seventeen  criminals  escaped  from  a  jail,  but  8  of 
them  were  caught.     How  many  secured  freedom  ? 

8.  A  boy  gathered  13  quarts  of  berries  and  sold  4 
quarts.     How  many  did  he  keep  ? 

9.  A  girl  who  had  20  cents  bought  nuts  for  8  cents. 
How  much  money  remained? 

10.  If  you  should  buy  a  cow  for  $20  and  sell  her  for 
$30,  how  much  would  you  gain  ? 


SUBTRACTION  83 

11.  How  much  would  I  lose  if  I  bought  a  wagon  for 
$45  and  sold  it  for  $25  ? 

12.  If  $1.00  is  the  minuend  and  75  cents  the  subtra- 
hend, what  is  the  remainder  ? 

13.  If  75  cents  is  the  subtrahend  and  25  cents  the  re- 
mainder, what  is  the  minuend  ? 

14.  What  is  the  second  principle  of  subtraction  ? 

15.  I  am  now  58  years  old.     How  old  was  I  15  years 
ago? 

16.  A  librarian  loaned  21  books;  all  were  novels  but  8. 
How  many  were  novels  ? 

17.  58  men  joined  the  army;  only  23  returned  home. 
How  many  were  unable  to  return  ? 

18.  The  United  States  flag  displays  45  stars ;  originally 
but  13.     What  has  been  the  increase  ? 

19.  How  many  are : 

1.  48  _  17  ?  8.  41  —  20  ?         15.  49  —  13  ? 

2.  39  —  18  ?  9.  36  —  16  ?         16.  44  —  33  ? 

3.  28  -      7  ?         10.  39  —    9  ?         17.  39  --  19  ? 

4.  59  —  19  ?         11.  48  —  11  ?         18.  48  —  18  ? 

5.  49  —  27  ?         12.  44  —  14  ?         19.  46  —  26  ? 

6.  42  —  12  ?         13.  38  —  14  ?         20.  39  —    9  ? 

7.  49  —  19  ?         14.  37  —  30  ?        21.  49  —  11  ? 

20.  Count  by  2's  from  50  to  2 ;  from  100  to  50. 

21.  Count  by  5's  from  100  to  60;  from  60  to  20. 

22.  Count  by  4's  from  100  to  40;  from  40  to  0. 

23.  Count  by  6's  from  90  to  60 ;  from  60  to  12. 

All  the  figures  of  the  minuend  may  be  of  greater 
value  than  the  corresponding  figures  of  the  subtrahend; 
and  one  or  more  figures  of  the  minuend  may  have  less 
value. 


84  ELEMENTARY  ARITHMETIC 

Fignres  of  the  Minuend  All  of  Greater  Value. 
WRITTEN   EXERCISES. 

1.  From  978  take  534. 

Process.  Explanation. 

Minuend       978  Since  the  unit  of  the  numbers  is  simply  one  with- 

o  VA    t.     j   co/i  out  name,  the  numbers  are  like  and  their  difference 

Subtrahend  5o4 

may  be  found. 

Remainder   444  We  write  units  of  the  same  order  in  the  same 

column,  and,  beginning  on  the  right,  say  8  units 

—  4  units  =  4  units ;  7  tens  —  3  tens  =  4  tens ;  9  hundreds  —  5  hun- 
dreds =  4  hundreds.     Hence,  the  remainder  is  444. 

Proof. 

444  +  534  =  978. 

Repeat  the  two  principles  of  subtraction. 
What  is  the  unit  of  978  horses  f     Of  53i  oxen  f 
Are  the  numbers,  then,  like  or  unlike  ? 
Can  their  difference  be  found  ? 

2.  Subtract  and  prove  the  following : 


(1) 

(2) 

(3) 

(4) 

(5) 

(6) 

(7) 

536 

972 

654 

890 

904 

529 

975 

325 

420 

313 

460 

402 

328 

342 

(8) 

(9) 

(10) 

(11) 

(12) 

(13) 

(14) 

968 

430 

900 

675 

999 

951 

938 

627 

220 

500 

324 

467 

511 

627 

(15)  (16)  (17)  (18)  (19)  (20) 

3976    9803    6975    9795     8396     9457 
2633    5603    3762    6254     5163     2346 


SUBTRACTION  85 

(21)                (22)                (23)                (24)  (25)  (26) 

$64.37       $59.40       $49.95       $59.99  $98.68  $99.87 

42.26         35.20         33.82         46.54  62.54  46.52 


(27) 
$93.27 

(28) 

$95.125 

(29) 

$98.96 

(30) 

$59.676 

(31) 

$635.89 

32.15 

24.125 

53.06 

49.250 

25.72 

"WRITTEN  PROBLEMS. 

1.  A  farm  is  worth  $896,  and  a  house  is  worth  $194. 
How  much  more  is  the  farm  worth  than  the  house  ? 

NOTE. — Indicate  all  processes,  thus  :  $896  —  $194  =  dollars  required. 

2.  A  boy  had  $1 .50 ;  he  bought  a  book  for  50  cents. 
How  much  money  had  he  left  ? 

3.  A  woman  went  to  market  with  a  5-dollar  bill ;  on 
her  return  she  had  a  2-dollar  bill  and  50  cents  change. 
How  much  had  she  spent  ? 

4.  A  man  bought   9893   bricks,  but  used  only  6473. 
How  many  had  he  left  ? 

5.  Gold  was  discovered  in  California  in  1848.     How 
long  ago  ? 

6.  Lucifer  matches  were  first  made  by  machinery  in 
1850.     How  long  ago  was  that? 

7.  I   bought   a    farm    for    $4796.30    and    sold    it    for 
$4897.50.     How  much  did  I  gain? 

8.  A  clerk  earned  in  a  year  $998.45  and  spent  $746.30. 
How  much  did  he  save  ? 

9.  A  brigade  going  into  battle  numbered  6975  men  ; 
after  the  battle  only  3844  responded  at  roll-call.     How 
many  had  the  battle  removed  ? 


86  ELEMENTARY  ARITHMETIC 

10.  Mr.  Hewitt  bought  a  city  lot  for  $2986  and  after- 
ward sold  it  for  $3986.     How  much  did  he  gain  ? 

11.  A  ship  that  cost  $7680  was  sold  for  $5650.     How 
great  was  the  loss  ? 

12.  A.'s  income  for  a  year  is  $9874.     How  much  does 
he  save  if  his  expenses  are  $4370  ? 

13.  Suppose  I  had  lent  a  man  1565  dollars  and  he  died 
owing  me  450  dollars.     How  much  had  he  paid  me  ? 

14.  A  person  sold  a  farm  for  15,896  dollars,  which  had 
cost  him  12,264  dollars.     How  much  did  he  gain? 

15.  Washington  died  in  1799,  at  the  age  of  67.     In 
what  year  was  he  born  ? 

16.  The  World's  Columbian  Exposition  at  Chicago  was 
held  in  1893.     How  many  years  after  the  discovery  of 
America  by  Columbus  ? 

17.  Lanterns  were  invented  by  King  Alfred  in  890. 
How  long  ago? 

18.  A    grocer    received    for    sugar    $308.40;    gained 
$106.20.     Find  the  cost. 

19.  How  much  nearer  is  1492  to  1792  than  to  1898  ? 

ORAL   EXERCISES. 
1.  Subtract  by: 

1.  5's  from  25,  saying  25,  20,  15,  etc. 

2.  10's  from  50  to  0.     From  100  to  0. 

3.  4's  from  20  to  0.     From  30  to  2. 

4.  3's  from  20  to  2.     From  30  to  0. 

5.  2's  from  20  to  0.     From  30  to  0. 

6.  6's  from  20  to  2.     From  30  to  0. 

7.  7's  from  20  to  6.     From  30  to  2. 

8.  8's  from  20  to  4.     From  30  to  6. 


SUBTRACTION  87 


9. 
10. 

9's  from  20  to  2. 
6's  from  52  to  4. 

From  30 
From  53 

to  3 
to  5 

• 

2.  How 

many 

are 

: 

I. 

25  — 

5? 

35  — 

5? 

45- 

5? 

55  — 

5? 

2. 

52  — 

4? 

62  — 

4? 

72  — 

4? 

82  — 

4? 

3. 

63  — 

6? 

73  — 

6? 

83  — 

6? 

93  — 

6? 

4. 

74  — 

8? 

84- 

8? 

94- 

8? 

24  — 

8? 

5. 

85  — 

9? 

95  — 

9? 

75  — 

9? 

65  — 

9? 

6. 

31  - 

2? 

41  - 

2? 

51  - 

2? 

61  - 

2? 

7. 

22  — 

3? 

32  — 

3? 

42- 

4? 

52  — 

5? 

8. 

44  — 

7? 

54  — 

7? 

64- 

7? 

74- 

7? 

9. 

39  — 

9? 

99  — 

9? 

109  —  9? 

110  —  9? 

10. 

33  — 

9? 

43  — 

8? 

53  — 

7? 

53  — 

8? 

Figures  of  Less  Value  in  the  Minuend. 
WRITTEN  EXERCISES. 

1.  From  110  subtract  9. 

• 

Process.  Explanation. 

110  HO  —  9  =  10*»  because  101  -j-  9  =  110;  but  we  may 

9  reason   thus,  saying:    "We   cannot  subtract  9  units   from 

T6T  °  units,  but  the  1  ten  =  10  units,  and  10  units  —  9  units  = 

1  unit,  which  we  write  in  the  remainder.     Having  used  the 

1  ten,  we  write  0  in  the  remainder,  and  also  the  1  hundred  which  we  did 

not  use.     Hence  110  —  9  =  101." 

The  following  is  the  extended  process  and  explanation  : 

110  =  1  hundred  -f  1  ten  -j-  0  units. 
9  =  9  units. 


By  reducing  the  1  ten  to  units,  the  above  becomes  : 

110  =  1  hundred  -f  0  tens  -f  10  units. 

9  =  9  units.    Subtracting,  we 

have  1  hundred  -j-  0  tens  -(-    1  unit  =  101. 


88  ELEMENTARY  ARITHMETIC 

2.  From  734  subtract  389. 
Process.  Explanation. 

Minuend       734  We  have   written   the  less    number  under  the 

Subtrahend  389  greater,  units  under  units,  and  tens  under  tens,  etc. 

3  and  4  of  the  minuend  are  of  less  value  than  8  and 

Remainder   345          9  of  the  subtrahend ;  9  cannot  be  taken  from  4. 
The  4  must  be  added  to  in  some  way.     One  of  the 

3  tens  =  10  units ;  10  units  -f  4  units  =  14  units ;  14  units  —  9  units 
=  5  units.  Instead  of  3  tens  we  have  now  but  2  tens  ;  8  cannot  be  taken 
from  2.  One  of  the  7  hundreds  =  10  tens ;  10  tens  -f  2  tens  =  12  tens ; 
12  tens  —  8  tens  =  4  tens.  Instead  of  7  hundreds  we  have  now  but 
6  hundreds ;  6  hundreds  —  3  hundreds  =  3  hundreds.  Hence,  the  re- 
mainder is  345. 

Proof. 
345  -f  389  =  734. 

The  extended  process  and  explanation  are  as  follows  : 

734  =  7  hundreds  -f  3  tens  -f  4  units. 

389  =  3  hundreds  -f  8  tens  -f  9  units. 

The  tens'  and  the  units'  figures  of  the  minuend  being  of  less  value  than 
the  tens'  and  units'  figures  of  the  subtrahend,  it  becomes  necessary  to  re- 
duce one  of  the  hundreds  to  tens,  and  one  of  the  tens  to  units.  By  doing 
this  we  have  734  =  6  hundreds  -f-  12  tens  -f  14  units. 

389  —  3  hundreds  -f-    8  tens  -|-    9  units.       Subtracting, 
we  have  3  hundreds  -(-    4  tens  -j-    5  units  =  345. 

61214  Instead  of  extending  the  work  as  above,  we 

may    make    the    reductions    and    indicate    the 

O  O  Q  J 

changes  by  placing  small  figures  above  the  fig- 
ures of  the  minuend,  as  in  the  margin. 

NOTE. — As  soon  as  pupils  understand  how  to  make  the  reductions, 
these  helps  should  be  omitted. 

To  THE  TEACHER.— Herein  lies  the  chief  difficulty  of  subtraction. 
Pupils  should  be  drilled  in  both  process  and  explanation  until  they 
thoroughly  understand  the  work  and  can  perform  it  with  facility.  Do 
not  pass  hurriedly  over  the  fundamental  operations ;  keep  them  before  the 
pupils  long  enough  to  make  an  impression.  Good  work  here  means  sure 
and  rapid  progress  hereafter.  Lay  a  good  foundation. 


SUBTRACTION  89 

On  what  principle  does  the  proof  depend? 

On  what  principle  does  the  above  process  depend  ? 

Brief  directions  are : 

1.  "Write  the  subtrahend  under  the  minuend,  units  under 
units,  tens  under  tens,  etc. 

2.  Begin  at  the  right  to  subtract. 

3.  Take  each  figure  of  the  subtrahend  from  the  figure 
above  it  in  the  minuend. 

4.  Add  1O  to  a  figure  of  less  value  in  the  minuend  and 
take  one  from  the  next  figure  on  the  left  in  the  minuend. 

5.  Prove  by  adding  the  remainder  to  the  subtrahend. 

3.  Subtract  and  prove  the  following : 


(1.)     (2.)     (3.)     (4.) 
546    792    695   725 
377    580    304   396 

(5.) 
938 
674 

(6.)     (7.)     (8.) 
592   493   827 
205   378   795 

(9.)     (10.)     (11.)    (12.) 

313    704    630   357 
247    195    548    249 

(13.) 

529 

483 

(14.)    (15.)    (16.) 

912   845   756 
345    678    680 

(17.)     (18.)     (19.)     (20.) 
$44.35  $35.29  $42.35  $49.42 

(21.)     (22. 

$34.92  $79. 

)    (23.) 
25  $94.37 

23.96 

27.37 

25.87 

28.37 

25. 

67   43. 

87   58.46 

(24.) 
$104.21 

(25.) 

$549.37 

(26) 

$345.93 

& 
(27.) 
$345.30 

(28.) 

$1786.08 

(29.) 

$3545.37 

75.80 

99.89 

76.04 

187 

.23 

1097.19 

966.38 

4.  From  3000  subtract  958. 

Process.  Explanation. 

3000  We  cannot  subtract  8  from  0.     We  therefore  take  1  ten 

958  from  the  300  tens,  leaving  299  tens.     1  ten  —  10  units  ;  10 

2042  units  —  8  units  =  2  units.     Instead  of  300  tens  we  have 

now   299   tens;    95   tens   from   299   tens   leaves    204    tens. 

Hence,  the  remainder  is  2042. 


90 


ELEMENTARY  ARITHMETIC 


Proof. 

2042  -f  958  =  3000. 

(1.)          (2.)          (3.)          (4.)  (5.)  (6.)  (7)  (8.) 

3507  4709  4000  5000  70000  80000  90000  70000 
2564  2678  2565  3794  34567  78910  12345  68904 


(9.)              (10.) 
100000     1000000 

(11.) 
1000 

(12.) 
$1000.00 

(13.)                  (14.) 

$234.56     $93000000 

99999 

1000 

407 

874.23 

87.678 

240000 

5.  Subtract: 

1.  404  from  795. 

2.  390  from  807. 

3.  567  from  954. 

4.  954  from  1005. 

5.  8376  from  10367. 

6.  4638  from  9524. 

7.  4076  from  5679. 

8.  4005  from  5967. 

9.  3786  from  6004. 
10.  4976  from  69005. 


11.  3679  from  38050. 

12.  40576  from  580676. 

13.  30794  from  490684. 

14.  430576  from  5600750. 

15.  767374  from  903727. 

16.  7503706  from  8732314. 

17.  758386  from  890573. 

18.  3885238  from  8630572. 

19.  8735918  from  9043057. 

20.  7953240  from  9932563. 


WRITTEN  PROBLEMS. 

1.  Find  the  difference  between  twenty  thousand  two 
and  twelve  thousand  eight  hundred. 

2.  Find  the  value  of  one  million  seven  hundred  less 
forty-mne  thousand  nine. 

3.  In  1892  a  city  had  58,376  inhabitants.    This  number 
was   greater  than   that  of  the   previous   year   by  5795. 
"What  was  the  number  of  the  previous  year  ? 


SUBTRACTION  91 

4.  A  man  bought  a  house  for  $7250  and  sold  it  at  a 
loss  of  $788.     How  much  did  he  sell  it  for  ? 

5.  A  man  has  $1980.     How  much  does  he  lack  of 
having  enough  to  buy  a  farm  worth  $13,000  ? 

6.  A  debtor  owing  $10,847,  paid  $4090.     How  much 
did  he  still  owe  ? 

7.  The  seeds  in  a  bushel  of  rye  number  888,390 ;  in  a 
bushel  of  wheat,  556,290.     Find  the  excess  of  rye  seeds. 

8.  How  much  must  be  added  to  109  to  make  1,000,000  ? 

9.  In  1895  the  production  of  wheat  in  Europe  was 
1,443,000,000  bushels;  in  the  United  States,  467,000,000 
bushels.     How  many  more   bushels  were   produced   in 
Europe  than  in  the  United  States  ? 

10.  In  1895  Iowa  produced  298,503,000  bushels  of  corn, 
exceeding  in  this  respect  any  other  State;  Montana,  in 
the  same  year,  33,000  bushels.     Find  the  difference. 

11.  In  1895  California  produced  40,098,000  bushels  of 
wheat;  Kansas,  29,919,000  bushels;  Pennsylvania  came 
third,  with  20,456,000  bushels.     How  far  did  California 
lead  each  of  the  two  other  States  ? 

12.  In  Hamilton  County,  Ohio,  the  internal  revenue 
tax  on  tobacco  in  1894  was  $1,375.32;  in  1895,  $1,170.36. 
Find  the  decrease. 

13.  In  Cincinnati  the  mean  temperature  of  summer  is 
76   degrees;   the   mean   temperature  of  winter   is   29.2 
degrees.     Find  the  difference. 

14.  Sound  travels  through  the  air  at  the  rate  of  1090 
feet  per  second  when  the  temperature  is  32  degrees,  and 
1  foot  faster  for  every  degree  the  temperature  rises.    How 
fast  per  second  does  sound  travel  when  the  temperature 
is  70  degrees  ? 


92  ELEMENTARY  ARITHMETIC 

ADDITION    AND    SUBTRACTION. 

ORAL   EXERCISES. 
How  many  are : 

1.4  +  3  —  5  +  4  +  6  +  0  —  3  —  5?     Say  4,  7,  2, 
6,  12,  9,  result  4. 

2.  7  —  1  +  4  —  5  +  7  —  3  —  8  +  9  —  4  +  5  +  8? 

3.  9  +  3  —  5  +  7  —  4  +  3  —  6  +  8  —  4  +  7  —  3? 

4.  6  +  5  —  3  —  4  +  7  —  3  +  4  —  5  +  7  —  6  +  8? 

5.  4  +  7  —  3  +  8  —  6  +  7  —  9  +  6  —  5  +  0  —  8? 

6.  9  +  6  —  5  +  7  —  4  —  3  +  7  +  5  —  5  —  4  +  5? 

7.  8  +  3  +  4  —  7  +  5  —  6  +  3  —  8  +  4  +  7  +  5? 

8.  9  +  3  —  7  —  4  +  7  —  3  +  5  —  6  +  7  —  7  +  3? 

9.  9  +  4  —  6  +  7  +  8  —  4  +  7  —  4  —  7  +  3  +  6? 

10.  9  +  3  —  7  +  4  —  3  —  2  +  9  —  5  —  4  +  7  +  3? 

11.  9  +  3  —  7  +  4  —  3  —  2  +  9  —  5  —  4  +  8  +  3? 

12.  10  —  4  +  8  —  5  —  6  +  4  +  5  —  3  —  6  +  5  + 

4  +  6? 

13.  11  —  1—2  —  3  —  4—1  +  1  +  2  +  3  +  4  + 

5  +  6? 

14.  12  —  2+3  —  4  +  5  —  6  +  7  —  8  +  9  —  0  + 

1  —  2  +  3? 

15.  i3_4  +  5_6  +  7  —  8  +  9  —  1  +  2  —  3  + 

4  — 5  +  6?  • 

16.  14  —  5  +  6  —  7  +  8  —  9  +  1—2  +  3  —  4  + 

5  —  6  +  7? 

17.  15  —  6  +  7  —  8  +  9  —  1  +  2  —  3  +  4  —  5  + 

6  —  7  +  8? 

18.  16  —  7  +  8  —  9  +  1—2  +  4  —  5  +  6  —  7  + 

8  —  9? 

19.  17  —  8  +  9  —  8  +  7  —  6  +  1  —  2  +  3  —  4  + 

5  —  6  +  7? 


SUBTRACTION  93 

WRITTEN   PROBLEMS. 

1.  Sarah  had  5  dollars  and  earned  10  more;  she  then 
spent  8  dollars.     How  much  money  had  she  left  ? 

2.  Margaret  wrote  12  letters  in  the  forenoon  and  14 
in  the  afternoon.     If  she  mailed  16  of  them,  how  many 
had  she  still  on  hand  ? 

3.  "William  owned  16  rabbits.     He  sold  4  to  Thomas, 
6  to  Henry,  and  bought  3  from  James.     How  many  rab- 
bits had  he  then  ? 

4.  On  leaving  Eighth  Street  a  trolley  car  had  6  pas- 
sengers ;  at  Fourteenth  Street  2  left  it  and  6  entered  it ; 
at  Twenty- second  Street  1  left  and  4  entered ;  at  Thirty- 
seventh  Street  4  more  entered;   at  Forty-second  Street 
2  left.     How  many  now  remained  ? 

5.  A  merchant  had  a  piece  of  cloth  containing  54 
yards.     He  sold  12  yards  to  one  man,  15  to  another,  and 
10  to  another.     How  many  yards  remained  in  the  piece  ? 

6.  A  gentleman  bought  a  watch  for  $70  and  a  chain 
for  $15.     He  sold  the  two  for  $97.     How  much  did  he 
gain  ? 

7.  A  farmer  had  70  sheep  in  one  field   and  65  in 
another.     He    sold    35    from    each    field.     How    many 
remained  ? 

8.  Virginia  bought  a  shawl  for  $16   and  a  pair  of 
gloves  for  $2.    She  gave  in  payment  two  $10  bills.     How 
much  change  was  due  her  ? 

9.  A  man  having  received  $56  for  his  services,  paid 
$25  for  a  coat,  $6  for  a  barrel  of  flour,  and  $5.50  for  a 
ton  of  coal.     How  much  money  had  he  left? 

10.  The  difference  of  the  ages  of  two  persons  was  49 


94  ELEMENTARY  ARITHMETIC 

years;  the  younger  person  was  born  in  1850.     In  what 
year  was  the  older  born  ? 

11.  James  has  97  cents;  he  pays  37  cents  for  a  whistle 
and  60  cents  for  a  knife.     How  much  has  he  left  ? 

12.  Two  men  start  from  the  same  place  and  travel  in 
the  same  direction.     When  one  has  travelled  65  miles 
and  the  other  47  miles,  how  far  will  they  be  apart? 

13.  John  had  31  marbles;  his  father  gave  him  9  more; 
he  sold  7,  found  4,  lost  8,  and  from  his  brother  received 
in  trade  4  for  5.     How  many  did  he  then  have  ? 

14.  A  farmer  paid  $35  for  a  cow,  $15  for  sheep,  $20  for 
pigs,  and  exchanged  them  all  for  a  hundred-and-fifty- 
dollar  horse.     How  much  did  he  have  to  pay  in  cash  ? 

15.  A  man  went  into  a  clothing-store  and  bought  a 
vest  for  $6.00,  a  coat  for  $20,  and  pantaloons  for  $11. 
He  handed  the  clerk  4  ten-dollar  bills.     How  much  did 
he  receive  in  change  ? 

WRITTEN   REVIEW. 

1.  What  is  an  equation  ? 

2.  Finish  the  following  equations  : 

1.  8975  +  4308  —  5904  +  3275  = 

2.  4927  —  3049  +  5574  —  5267  = 

3.  9240  +  3796  —  5432  —  5237  = 

4.  24570  +  27957  —  4907  +  3275  = 

5.  37920  —  5970  —  24357  +  3795  = 

6.  46249  —  10987  —  9854  —  24523  = 

7.  4062  +  12356  +  15000  —  975  = 

8.  23462  +  9030  +  34000  —  7640  = 

9.  19876  —  6032  —  12000  +  673  = 

10.  87642  +  798764  —  379862  —  4001  —  14760  = 


SUBTRACTION  95 

3.  Find  the  difference  between  twenty  thousand  two 
and  twelve  thousand  eight  hundred. 

4.  A  man  has  $10,000.     How  much  must  he  borrow 
that  he  may  be  able  to  pay  $24,750  for  an  estate  ? 

5.  Three  men  bought  a  building  for  $47,956.     If  the 
first  paid  $21,706,  and  the  second  $9575,  how  much  re- 
mained for  the  third  to  pay  ? 

6.  If  I   deposit  in   one   bank   $1095.54,  in   another 
$987.95,  and  in  a  third  $709.28,  how  much  must  I  bor- 
row to  enable  me  to  build  a  house  worth  $5486  ? 

7.  The  sum  of  three  numbers  equals  98,765;  two  of 
the  numbers  are  28,907  and  36,794;  what  is  the  third 
number  ? 

8.  A  man  sold  a  firkin  of  butter  for  $20,  a  cheese  for 
$10,  and  a  quantity  of  fruit  for  $12.50.     He  received  in 
payment  5  barrels  of  flour  valued  at  $25.75.     How  much 
is  still  owing  to  him  ? 

9.  A  farmer,  having  625  bushels  of  grain,  sold  to  A. 
97  bushels,  to  B.  127,  to  C.  197,  and  gave  110  bushels  to 
the  poor.     How  many  bushels  had  he  remaining  ? 

10.  A  man  deposited  $10,000  in  a  bank.     He  drew  out 
at  one  time  $47.00 ;  at  another,  $24.80 ;  and  at  another, 
$1474.     How  much  had  he  remaining  in  the  bank? 

11.  A  boy  bought  a  sled  for  $2.60,  and  gave  $1.00  for 
having  it  repaired.     He  sold  it  for  $4.00,  and  lost  $2.50 
of  the   money.     To  what   extent  was   he   then    out   of 
pocket  ? 

12.  I  bought  28  yards  of  cloth  for  $26.54,  10  yards  of 
calico  for  $2.72,  25  pounds  of  sugar  for  $1.375,  4  pounds 
of  tea  for  $1.95.     I  paid  $9.85.     How  much  do  I  still 
owe? 


96  ELEMENTARY  ARITHMETIC 

13.  A.  had  450  sheep,  B.  had  175  more  than  A.,  and 
C.  had  as  many  as  A.  and  B.  together  minus  114.     How 
many  sheep  had  C.  ? 

14.  Mr.  Swift  has  in  cash  $419.14,  hut  he  owes  Mr. 
Brick  $47.55,  Mr.  Quick  $274.30,  and  Mr.  Lick  $97.29. 
After  paying  these  gentlemen,  how  much  will  he  have 
left? 

15.  A.  paid  $9000  for  his  farm,  $8476  for  a  new  house, 
and  $873  for  a  barn,  and  then  sold  them  for  $29,600. 
What  was  his  gain  ? 

16.  In  1890  the  exports  of  the  United  States  amounted 
to    $857,828,684,   the    imports    to    $789,310,409.      How 
much  does  the  sum  of  these  two  amounts  exceed  their 
difference  ? 

17.  A  farmer  had  738  sheep.     He  sold  327  to  A.  and 
234   to   B.      Disease    carried   off   90.      How  many   re- 
mained ? 

18.  I  paid  $2240  for  some  land.     I  sold  coal  for  $116, 
a  horse  for  $225,  a  wagon  with  a  horse  attached  for  $277, 
and  collected  a  bill  of  $565.25  long  due.     How  much 
more  did  I  pay  out  than  I  received  ? 

19.  If  February  has  28  days,  and  April,  June,  Sep- 
tember  and   November   have   120  days,  the   remaining 
seven  months  supply  how  many  days  to  make  up  365 
days? 

20.  If  June  (30  days),  July  (31  days),  and  August  (31 
days)  are  the  summer  months,  and  December  (31  days), 
January  (31  days),  and  February  are  the  winter  months, 
which  is  the  longer,  summer  or  winter,  and  how  much  ? 

21.  I  start  out  with  $205.50  in  one  pocket  and  $43.25 
in  the  other.    I  pay  the  grocer  $63.69,  the  butcher  $32.08 


SUBTRACTION  97 

the  shoemaker  $10.70,  the   landlord   $37.50,  the  tailor 
$17.50.     How  much  have  I  left  ? 

22.  Margaret  has  74  cents,  Mary  has  135  cents.     Mary 
gave  Margaret  28  cents.     Which  has  then  the  greater 
sum,  and  greater  by  how  much? 

23.  If  the  minuend   is   $4937  and   the  remainder  is 
$1593,  what  is  the  subtrahend?     The  subtrahend  being 
$3825  and  the  remainder  $337.84,  what  is  the  minuend  ? 

24.  From  what  number  must  I  subtract  10  to  leave 
14  ?     16  to  leave  18  ?     2.586  to  leave  4.098  ? 

25.  What  sum  must  be  subtracted  from  $1.00  to  leave 
$.50?      To  leave   $.625?      To  leave  $.375?     To  leave 
$.875? 

26.  What  number  increased  by  63,915  makes  a  million  ? 

27.  If  in   July   220,860   carriages    visited   Fairmount 
Park,  5575  equestrians,  and  1,443,173  pedestrians,  find 
the  excess  of  pedestrians  over  the  other  two  classes. 

28.  A  lady  bought  a  bonnet  for  $12,  a  pair  of  shoes  for 
$4,  and  a  fan  for  $1.     She  gave  the  salesman  a  $20  bill. 
What  change  did  she  receive  ? 

29.  Find  the  value  of  MCLX.  +  LXXXVIII.  —  DXL. 
+  IX.  +  IV.  —  XC. 

30.  Find  the  value  of  MCMXCYIII.  —  XCVIII.  — 
DCCC.  —  M. 

31.  A  man  had  in  bank  $15,000,  deposited  $3875,  drew 
out  $8725,  and  then  put  in  enough  to  make  his  deposit 
$20,000.     How  much  did  he  last  put  in  ? 

32.  $15,000  is  divided  among  three  children,  the  second 
of  whom  receives  $2240  more  than  the  first,  and  the  third 
of  whom  receives  the  remainder.     If  the  first  receives 
$5000,  how  much  does  each  of  the  others  receive  ? 

7 


98 


ELEMENTARY  ARITHMETIC 


MULTIPLICATION. 

INDUCTIVE  STEPS. 


1.  How  many  are  : 

1.  1  +  1  ?     4  +  4  ? 

2.  2  +  2  ?     5  +  5  ? 

3.  3  +  3  ?     6  +  6  ? 

2.  Hence,  we  may  say  : 

1.  Two  times  1  =  2. 

2.  Two  times  2  =  4. 

3.  Two  times  3  =  6. 

3.  Instead  of  writing : 

2.  2  +  2  +  2  =  6. 

3.  3  +  3  +  3  =  9. 

We  can  write  more  briefly : 


Two  times  4  =  8. 
Two  times  5  =  10. 
Two  times  6  =  12. 


4  +  4  +  4  =  12. 

5  +  5  +  5  =  15. 

6  +  6  +  6  =  18,  etc. 


1.  Three  times  1=3. 

2.  Three  times  2  =  6. 

3.  Three  times  3  =  9. 


Three  times  4  =  12. 
Three  times  5  =  15. 
Three  times  6  —  18,  etc. 


2  x  1  =  2  is  read  "two  times  one  equal  two";  the 
word  times  being  expressed  by  the  sign. 

The  process  of  thus  shortening  addition  is  called  Mul- 
tiplication. 

4.  Proceeding  in  like  manner  with  4,  5,  6,  7,  8,  9,  10, 
11,  12,  and  13,  we  construct  the  following  table : 


MULTIPLICATION 


99 


Multiplication  Table. 


2X1=  2 

3X1=   3 

4X  1=   4 

5x  1=   5 

2X2=4 

3X2=6 

4X2=8 

5  X  2  =  10 

2X  8=   6 

3X3=   9 

4  X  3  =  12 

5  X  3  =  15 

2X4=8 

3  X  4  =  12 

4  X  4  =  16 

5  X  4  =  20 

2  X  5  =  10 

3  X  5=  15 

4  X  5  =  20 

5X  5=  25 

2X  6=  12 

3  X  6  =  18 

4  X  6  =  24 

5  X  6  =  30 

2X  7=  14 

3  X  7  =  21 

4  X  7  =  28 

5X  7=  35 

2  x  8  =  16 

3  X  8  =  24 

4  X  8  =  32 

5  X  8  =  40 

2  X  9—  18 

3  X  9  =  27 

4  X  9  =  36 

5  X  9  =  45 

2  X  10  =  20 

3  X  10  =  30 

4  X  10  =  40 

5  X  10  =  50 

2  X  11  =  22 

3  X  11  =  33 

4  X  11  =  44 

5  X  11  =  55 

2  X  12  =  24 

3  X  12  =  36 

4  X  12  =  48 

5  X  12  =  60 

6X1=   6 

7X1=   7 

8X1=   8 

9X1=   9 

6  X  2  =  12 

7  X  2  =  14 

8  X  2  =  16 

9  X  2  =  18 

6  X  3  =  18 

7  X  3  =  21 

8  X  3  =  24 

9  X  3  =  27 

6  X  4  =  24 

7  X  4  =  28 

8  X  4  =  32 

9  X  4  =  36 

6  X  6  =  30 

7  X  5  =  35 

8  X  5  =  40 

9  X  5  =  45 

6  X  6  =  36 

7  X  6  =  42 

8  X  6  =  48 

d  X  6  =  54 

6  X  7  =  42 

7  X  7  =  49 

8  X  7  =  56 

9  X  7  =  63 

6  X  8  =  48 

7  X  8  =  56 

8  X  8  =  64 

9  X  8  =  72 

6  X  9  =  54 

7  X  9  =  63 

8  X  9  =  72 

9  x  9  =  81 

6  X  10  =  60 

7  X  10  =  70 

8  X  10  =  80 

9  X  10  =  90 

6  X  11  =  66 

7  X  11  =  77 

8  X  11  =  88 

9  X  11  =  99 

6  X  12  ==  72 

7  X  12  =  84 

8  X  12  =  96 

9  X  12  =  108 

10  X  1  =  10 

11  X  1=  11 

12  X  1  =  12 

13  X  1  =  13 

10  X  2  =  20 

11  X  2  =  22 

12  X  2  =  24 

13  X  2  =  26 

10  X  3  =  30 

11  X  3  =  33 

12  X  3  =  36 

13  X  3  =  39 

10  X  4  ==  40 

11  X  4  =  44 

12  X  4  =  48 

13  X  4  =  52 

10  X  6  =  50 

11  X  5  =  55 

12  X  5  =  60 

13  X  6  =  65 

10  X  6  =  60 

11  X  6  =  66 

12  X  6  =  72 

13  X  6  =  78 

10  X  7  =  70 

11  X  7  =  77 

12  X  7  =  84 

13  X  7=  91 

10  X  8  =  80 

11  X  8  =  88 

12  x  8  =  96 

13  X  8  =  104 

10  x  9  =  90 

11  X  9  =  99 

12  X  9  =  108 

13  X  9  =  117 

10  X  10  =  100 

11  X  10  =  110 

12  X  10  =  120 

13  X  10  =  130 

10  X  11  =110 

11  X  11  =  121 

12  X  11  =  132 

13  X  11  =  143 

10  X  12  =  120 

11  X  12  =  132 

12  X  12  =  144 

13  X  12  =  156 

100  ELEMENTARY  ARITHMETIC 

5.  In  the  column  beginning  with  4  X  1  =  4,  show  how 
all  the  results,  4,  8,  12,  16,  etc.,  were  obtained. 

6.  The   multiplication   table   is   an    invention   of  the 
greatest  value.     If  you  do  not  yet  know  this  table,  learn 
it  perfectly  as  soon  as  possible. 

DEFINITIONS. 

1.  Multiplication  is  a  short  process  of  adding  when 
the  numbers  to  be  added  are  all  equal;  or,  it  is  the  pro- 
cess of  finding  the  result  of  repeating  one  number  as 
many  titnes  as  there  are  units  in  another. 

2.  The  Multiplicand  is  the  number  to  be  repeated  or 
multiplied. 

3.  The  Multiplier  is  the  number  by  which  we  mul- 
tiply. 

4.  The  Product  is  the  result  of  multiplying. 

5.  The  Multiplicand  and  Multiplier  are  called  Factors 
of  the  product. 

6.  We  may  write  5   times  12  equal  60,  or  we  may 
write  5  times  12  men  equal  60  men,  or  five  times  $12 
equal  $60.     In  the  first  case,  12  and  60,  having  no  appli- 
cation to  men,  or  money,  or   other  things,  are  called 
Abstract  Numbers ;  in  the  second  case,  being  applied  to 
men,  and  in  the  third  case  to  dollars,  12  and  60  are  called 
Denominate  Numbers. 

The  multiplier  5  is  abstract  in  all  the  cases;  show- 
ing no  more  than  the  number  of  times  12  is  to  be 
taken. 

7.  Since  5  X  12  =  60,  and  12  X  5  =  60,  we  see  that 
the  product  is  the  same  whether  5  or  12  is  used  as  the 
multiplier. 


MULTIPLICATION  101 


PRINCIPLES. 

1.  The  multiplier  must  be  used  as  an  abstract 
number. 

2.  The   multiplicand   and  the   product   are   like 
numbers. 

3.  Either  factor,  "when  abstract,  may  be  used  as 
multiplier  or  multiplicand. 


ORAL    EXERCISES. 

1.  Multiply  9679765896948 
By  548609794869. 6 

2.  Multiply  7865876585689 
By  ^479674848756 

3.  Multiply  0234567891234 
By  _9_?_?65^78898789 

4.  Multiply  10    11    12    13    9    10    8    11    7    12    6    13 
By  _5    Jt  _6  _7    8  _910    U12  _7    6  _5 

Is8X9  =  9X8a  true  equation  f 

By  what  name  are  8  and  9  called  in  their  relation  to  the 
product  ? 

ORAL   PROBLEMS. 

1.  A  dime  equals  10  cents.     8  dimes  equal  how  many 
cents  ? 

2.  If  6  hours  make  a  school  day,  how  many  hours  in  5 
school  days  ? 

•  3.  How  much  will  2  pencils  cost  at  5  cents  each  ? 


102  ELEMENTARY  ARITHMETIC 

4.  There  are  7  days  in  a  week.     How  many  days  in  6 
weeks  ? 

5.  How  many  are  9  times  5  pears  ?     5  times  9  pears  ? 

6.  How  much   can  a  man   earn   in  11   days  at  $5  a 
day? 

7.  How  mueh  will  3  cows  cost  at  $13  each  ? 

8.  What  is  the  cost  of  2  oranges  at  5  cents  each  and 
of  4  bananas  at  3  cents  each  ? 

9.  How  many  days  in  12  weeks? 

10.  How  much  must  I  pay  for  9  sheep  at  $7  a  head  ? 

11.  If  5  men  can  do  a  piece  of  work  in  3  days,  how 
long  will  it  take  1  man  ? 

12.  If  12  men  can  do  a  piece  of  work  in  10  days,  how 
many  men  will  be  required  to  do  it  in  1  day  ? 

13.  How  many  are  6  X  4  +  7  ?     8  X  5  +  30  ? 

14.  How  many  are  5  X  6  +  10  —  25  ? 

15.  If  a  boy  earns  $7  a  month,  how  much  can  he  earn 
in  9  months  ? 

16.  At  2  cents  a  foot,  how  much  will  13  feet  of  board 
cost? 

17.  Repeat  the  first,  second,  third,  and  fourth  columns 
of  the  multiplication  table. 

18.  Repeat  the  fifth,  sixth,  seventh,  and  eighth  columns 
of  the  table. 

19.  Repeat   the    ninth,   tenth,   eleventh,   and   twelfth 
columns  of  the  table. 

20.  Declare  promptly  the  product  of: 

1.  2  X  11.  5.  5  X  8.  9.  8  X  4.  13.  10  X  9. 

2.  3  X  10.  6.  6  X  7.  10.  9x3.  14.  11  X  11. 

3.  4  X  9.  7.  7  X  5.  11.  7x7.  15.  12  x  6. 

4.  4  X  7.  8.  5  X  9-  12.  9x9.  16.  2  X  12.* 


MULTIPLICATION  103 

The  Multiplier  a  Single  Figure. 

"WRITTEN   EXERCISES. 

1.  How  many  are  5  times  1728  ? 

Process.  Explanation. 

Multiplicand,  1728  )  !•  5  *imes  8  units  =  40  unite  = 

V  Factors.  4  tens  +  0  units. 

Multiplier.  5  J 

2.  5  times  2  tens  =  10  tens ;  10 

Produ0t'  tens  +  4  tens  =  14  tens  =  1  hun- 

dred  -f-  4  tens. 

3.  5  times  7  hundreds  =  35  hundreds  ;  35  hundreds  -f  1  hundred  •=  36 
hundreds  =  3  thousands  -|-  6  hundreds. 

4.  5  times  1  thousand  =  5  thousands ;  5  thousands  -f  3  thousands  = 
8  thousands. 

Hence,  the  product  is  8640. 

'Proof. 

1728  +  1728  +  1728  +  1728  -f  1728  =  8640. 

2.  Find  the  product  of  the  following  factors  : 

1.  347  X  5.  12.  6038  X  7.  23.  42598  X  8. 

2.  357  X  6.  13.  7984  X  9.  24.  30487  X  7. 

3.  530  X  4.  14.  6346  X  8.  25.  123456  X  2. 

4.  937  X  4.  15.  5396  X  9.  26.  789012  X  3. 

5.  348  X  5.  16.  3140  X  7.  27.  345678  X  4. 

6.  905  X  3.  17.  53645  X  9-  28.  901234  X  5. 

7.  570  X  6.  18.  30724  X  6.  29.  567890  x  6. 

8.  943  X  5.  19.  42304  X  9.  30.  123456  X  7. 

9.  3072  x  4.  20.  53728  X  7.  31.  890123  X  8. 

10.  5937  X  7.   21.  43954  X  9.  32.  456789  X  9. 

11.  8945  x  6.   22.  31708  X  8.  33.  876543  X  5. 


104 


ELEMENTARY  ARITHMETIC 


"WRITTEN   PROBLEMS. 

1.  What  will  9  pairs  of  shoes  cost  at  $3.50  a  pair  ? 

2.  At  $25  each,  what  will  8  cows  cost  ? 

3.  At  $.50  each,  what  will  5  books  cost? 

4.  In  one  square  foot  there  are  144  square  inches. 
How  many  square  inches  in  6  square  feet  ? 

5.  There  are  365  days  in  a  year.     How  many  days  in 
8  years  ? 

6.  If  sound  travels  1120  feet  in  a  second,  how  far  does 
it  travel  in  9  seconds  ? 

7.  In  one  bushel  there  are  2150.42  cubic  inches.    How 
many  cubic  inches  in  7  bushels  ? 

8.  There  are  5280  feet  in  a  mile.     How  many  feet  in 
6  miles  ? 

9.  What  is  the  value  of: 

1.  8  X  5  +  6  +  10?        6.  0  X  7  +  8  +  8? 

Suggestion  :  Multiply  before  adding  or  subtracting. 


2.  5X9  —  5X6? 

3.  60  —  6  X  1  +  8  ? 

4.  9X5  —  7X1? 

5.  5X9  +  4  +  6? 
10.  Find 

The  sum  of:         The  product  of : 

57837  1.  7^9127  X  6. 

786584  2.  832561  X  7. 

828763  3.  563656  X  8. 

683784  4.  624526  X  9. 

148528  5.  536465  X  7. 

691789  6.  563464  X  8. 

979897  7.  399978  X  6. 


7.  13X6  +  4  +  8X2? 

8.  10  X  8  +  10  — 7? 

9.  39  —  30  +  5X9? 
10.  73  +  8—9x9? 

The  difference  of: 

1.  7000305  and  663650. 

2.  9043057  and  873598. 

3.  9540371  and  7936597. 

4.  7987396  and  6798309. 

5.  7799458  and  5988799. 

6.  8735009  and  7295394. 

7.  6937254  and  2786927. 


MULTIPLICATION  105 

The  Multiplier  with  Ciphers  Annexed. 
INDUCTIVE    STEPS. 

1.  What  is  the  product  of  the  following  factors? 

1.  5  and  10?     6  and  10?     7  and  10? 

2.  10  and  5  ?     10  and  6  ?     10  and  7  ? 

The  significant  figure  in  each   product  has  what  an- 
nexed to  it  ? 

Annexing  a  cipher,  then,  multiplies  by  what  ? 

2.  8  times  100  =  what?     100  times  8  =  what? 

The  significant  figure  8  has  how  many  ciphers  annexed 
in  the  product  ? 

3.  9  X  1000  =  what?     1000  X  9  =  what? 

The  significant  figure  9  in  the  product  has  how  many 
ciphers  annexed  ? 


PRINCIPLES. 

1.  Annexing  1  cipher  multiplies  by  1O. 

2.  Annexing  2  ciphers  multiplies  by  1OO. 

3.  Annexing  3  ciphers  multiplies  by  1OOO,  and 
so  on. 


"WRITTEN   EXERCISES. 
1.  Multiply  144  by  10. 

Process.  Explanation. 

Multiplicand,  144     1    Factorg  '  In  accordance  with  Principle  1, 

i  f\  I  to  multiply  by  ten.  we  annex  one 

Multiplier,  1U  J 

cipher  to  the  multiplicand,  and  ob- 
Product,  tain  for  product  1440. 

Or,    we   may   say:     "144   times 

1  ten  =  144  tens ;   but  144  tens  =  1440  units.     Hence,  the  product  is 
1440." 


106  ELEMENTARY  ARITHMETIC 

2.  Multiply  365  by  1000. 

Process.  Explanation. 

365          \  In  accordance  with  Principle  3,  to  multiply 

1000  /  kj  100°  we  annex  three  ciphers  to  the  multi- 

*}fiW)0  pHcand,  and  obtain  for  product  365,000. 

Or,  we  may  say  :  "  365  times  1  thousand  = 
365  thousands  ;   but   365   thousands  =  365,000 
units.     Hence,  the  product  is  365,000." 

3.  Multiply  1728  by  3000. 

Process.  Explanation. 

1728  3000  =  3  times  1000;    1728   multiplied  by  1000  = 

3000  1,728,000;  1728  multiplied  by  3000  =  3  times  1,728,000, 

— — — —  or  5,184,000. 

Or,  we  may  say:    "1728  times  3  thousands  =  5184 
thousands  ;  but  5184  thousands  =  5,184,000  units.    Hence, 
the  product  is  5,184,000." 

RULE. 

1.  Cut  off  the  ciphers  from  the  multiplier. 

2.  Multiply  by  the  significant  figure. 

3.  To  the  product  annex  the  ciphers  cut  off. 
To  apply  the  rule  : 

4.  Multiply  1898  by  400. 

Process.  Explanation. 

1898  Having  cut  off  the  two  ciphers,  we  then  multiply  1898 

4|00  ^  *  an<^  °^tam   7592.     Annexing  the  two  ciphers,  we 


759200 

nave  /oy,zut 

r. 

5.  Multiply: 

(1) 

(2) 

(3), 

(4) 

(5) 

579 

480 

269 

3760 

3405 

100 

1000 

100 

1000 

100 

(6) 

(7) 

(8) 

(9) 

(10) 

4500 

4000 

5700 

779 

495 

100 

1000 

1000 

100 

30 

MULTIPLICATION  107 


(11) 

570 

(12) 
385 

(13) 
679 

(14) 
607 

(15) 
508 

40 

60 

80 

300 

400 

(16) 

679 

(17) 
572 

(18) 
934 

(19) 

768 

(20) 

938 

800 

300 

700 

4000 

5000 

(21) 
1136 

(22) 
922 

(23) 

1646 

(24) 

1336 

(25) 

2672 

200 

50 

6000 

70 

9000 

(26) 
548 

(27) 
1136 

(28) 

592 

(29) 
1192 

(30) 

994 

6000 

7000 

8000  x 

9000 

1000 

The  Multiplier  Two  or  More  Significant  Figures. 

1.  Multiply  374  by  243. 

Process.  Explanation. 

375  We  write  the  multiplier  under  the  multiplicand,  units 

04*5  under  units,  etc. 

375  multiplied  by  3  units  =  1125  units.     (Why  ?) 
375  multiplied  by  4  tens  =  1500  tens.     (Why  ?) 
375  multiplied  by  2  hundreds  =  750  hundreds.    (Why  ?) 
Writing  these  three  products  in  order  as  units,  tens  and 
hundreds,  and  adding,  we  have  as  total  product  91,125. 

2.  Multiply  3703  by  408. 

Process.  Explanation. 

3703  8  times  3703  =  29,624  units. 

4Q£  0  times  3703  =  0  tens. 

4  times  3703  =  14,812  hundreds. 
Adding  the  partial  products,  we  have  1,510,824. 
148120 

1510824 


108          ELEMENTARY  ARITHMETIC 

3.  Multiply: 

1.  370  by  43.  31.  30796  by  245. 

2.  596  by  35.  32.  49075  by  406. 

3.  704  by  45.  33.  30986  by  570. 

4.  836  by  54.  34.  39687  by  542. 

5.  937  by  63.  35.  30425  by  341. 

6.  938  by  46.  36.  37968  by  535. 

7.  937  by  35.  37.  79846  by  270. 

8.  925  by  66.  38.  57964  by  329. 

9.  530  by  74.  39.  46854  by  213. 

10.  739  by  59.  40.  67895  by  304. 

11.  832  by  68.  41.  579656  by  2134. 

12.  740  by  86.  42.  678956  by  3045. 

13.  7954  by  78.  43.  897362  by  4606. 

14.  9836  by  85.  44.  812357  by  4328. 

15.  9730  by  94.  45.  183586  by  2345. 

16.  2397  by  99.  46.  970890  by  7980. 

17.  3805  by  70.  47.  759671  by  9087. 

18.  3496  by  68.  48.  935802  by  9358. 

19.  3795  by  47.  49.  215090  by  7809. 

20.  5368  by  72.  50.  195833  by  9800. 

21.  $46.96  by  46.  51.  370184  by  10708. 

22.  $27.30  by  78.  52.  508199  by  25409. 

23.  $35.29  by  40.  53.  190199  by  89760. 

24.  $35.46  by  54.  54.  280755  by  20499. 

25.  $34.10  by  76.  55.  162788  by  72644. 

26.  $59.42  by  79.  56.  320988  by  23577. 

27.  $29.68  by  80.  57.  684579  by  323500. 

28.  $38.96  by  43.  58.  123456  by  789000. 

29.  $29.875  by  50.  59.  987643  by  210034. 

30.  $68.57  by  65  60.  567890  by  123456. 


MULTIPLICATION  109 

WRITTEN   PROBLEMS. 

1.  In  one  barrel  of  flour  there  are  196  pounds.     How 
many  pounds  in  434  barrels  ? 

2.  A  railway  train  travels  an  average  of  235  miles 
per  day.     How  far  does  it  travel  in  31  days  ? 

3.  There  are  1760  yards   in  one  mile.     How  many 
yards  in  397  miles  ? 

4.  If  a  man  travels  30  miles  in  one  day,  how  many 
miles  will  he  travel  in  49  days  ? 

5.  A  farmer  raised  1448  bushels  of  wheat,  which  he 
sold  at  76  cents  per  bushel.     How  much  did  he  receive 
for  it? 

6.  I  sold  my  farm  of  302  acres  at  $105  per  acre.     How 
much  did  I  get  for  it  ? 

7.  Find  the  cost  of  759  articles  costing  $5.77  each. 

8.  Find  the  cost  of  367  things  at  $23.34  each. 

9.  There  are  5280  feet  in  a  mile.     How  many  feet  are 
there  in  17  miles  ? 

10.  How  many  oranges  are  there  in  47  boxes,  if  each 
box  contains  189  oranges? 

1 1 .  How  many  pounds  of  hay  will  299  acres  produce, 
if  each  acre  produces  2178  pounds. 

12.  There  are  4840  square   yards  in  an  acre.      How 
many  square  yards  in  127  acres? 

13.  If  a  state  has  an  area  of  8040  square  miles,  and  a 
population  of  279  to  the  square  mile,  what  is  the  popula- 
tion of  the  state  ? 

14.  A  merchant   purchased  30  pieces  of  broadcloth, 
each  containing  50  yards,  at  $8  per  yard.     How  much 
did  he  pay  for  the  whole  ? 


HO  ELEMENTARY  ARITHMETIC 

15.  I  sold  27  loads  of  wheat,  35  bushels  in  each,  at  93 
cents  a  bushel.     How  much  money  did  I  receive  ? 

ORAL    REVIEW. 

1.  What  is  the  cost  of  2  oranges  at  5  cents  each  and 
4  bananas  at  3  cents  each  ? 

2.  I  bought  2  slates  at  15  cents  apiece  and  3  pencils  at 

4  cents  each.     How  much  did  I  pay  for  all  ? 

3.  I  bought  3  pencils  at  4  cents  each  and  4  oranges  at 

5  cents  each.     How  much  did  I  pay  for  all  ? 

4.  If  a  man  earns  $6  per  week  and  a  boy  earns  $3 
per  week,  how  much  more  will  the  man  earn  in  12  weeks 
than  the  boy  ? 

5.  A  farmer  has  30  bags  of  wheat,  each  containing  3 
bushels.     How  much  is  the  wheat  worth  at  $1.00  per 
bushel  ? 

6.  If  6  teams  can  plough  a  field  in  4  days,  how  long 
will  it  take  one  team  to  do  the  same  work  ? 

7.  William  has  25  cents  and  Harry  has  4  times  as 
many  plus  10.     How  many  has  Harry  ? 

8.  A  man  bought  a  plough  for  $13  and  3  harrows  at 
$9  apiece.     How  much  did  he  pay  for  all  ? 

9.  How  much  must  I  pay  for  3  books  at  30  cents  each 
and  5  at  25  cents  each  ? 

10.  A  matron  purchased  4  bunches  of  asparagus  at  10 
cents  a  bunch  and  5  quarts  of  strawberries  at  13  cents 
each.     How  much  did  she  pay  for  all  ? 

11.  George  earned  95  cents  a  day;  his  board  cost  him 
55  cents  a  day.     How  much  did  he  save  in  20  days  ? 

12.  A  gallon  contains  4  quarts,  and  a  quart  2  pints. 
How  many  pints  in  6  gallons  ? 


MULTIPLICATION  111 

13.  A  drover  bought  50  sheep  for  120  dollars;  he  sold 
20  of  them  at  5  dollars  a  head  and  the  rest  at  4  dollars 
a  head.     How  much  did  he  gain  ? 

14.  I  bought  4  quarts  of  milk  at  8  cents  a  quart  and 
4  pounds  of  biscuits  at  10  cents  a  pound.     If  I  gave 
a  dollar  note  in  payment,  how  much  should  I  receive 
back? 

15.  In  an  orchard  are  26  peach-trees  and  6  times  as 
many  apple-trees  as  peach-trees.     How  many  trees  in  the 
orchard  ? 

16.  At  4.5  cents  a  pound,  what  will  320  pounds  of 
sugar  cost? 

17.  If  20  pennyweight  make  an  ounce,  what  will  a 
2-ounce  gold  chain  cost  at  $1.25  a  pennyweight? 

18.  How  much  are  5  loads  of  flour  worth,  each  load 
containing  12  barrels,  at  $6.00  a  barrel? 

19.  A  farmer's  wife  took  to  a  store  6  pounds  of  butter 
at  30  cents  a  pound  and  bought  15  yards  of  calico  at  10 
cents  a  yard.     Find  the  balance  due  her  ? 

20.  Mr.  Green  owns  three  houses;  the  first  is  worth 
$12,000,  the  second  is  worth  twice  as  much  as  the  first, 
and*  the  third  as  much  as  both  the  first  and  second.    How 
much  is  each  house  worth,  and  how  much  are  they  all 
together  worth  ? 

WRITTEN    REVIEW. 

1.  Add  6  thousand  4  hundred  80,  4  thousand  4,  8 
thousand  7,  8  hundred  89,  54  hundred  11. 

2.  A  windmill  pumps  into  a  tank  25  gallons  of  water 
in  an  hour.     After  pumping  13  hours,  how  much  does 
the  tank,  holding  500  gallons,  lack  of  being  filled  ? 


112  ELEMENTARY  ARITHMETIC 

3.  A  merchant  bought  240  barrels  of  flour  for  $1920 
and  sold  it  at  $10.50  a  barrel.     What  did  he  gain  ? 

4.  Which  cost  the  more  and  how  much, — 48  horses  at 
$173.40  each  or  1130  sheep  at  $4.60  a  head? 

5.  If  1 69  tons  of  steel  rails  are  required  for  a  mile  of 
railroad,  how  many  tons  are  required  for  449  miles  ? 

6.  A  farmer  bought  40  sheep  at  $7.00  each,  40  cows 
at  $30  per  head,  and  4  horses  at  $200  each ;  he  sold  them 
all  for  $3000.     How  much  did  he  gain  ? 

7.  What  cost  37,560  tons  of  iron  at  $57  per  ton  ? 

8.  What  cost  293  barrels  of  flour  at  $7  per  barrel, 
and  729  barrels  at  $8  per  barrel  ? 

9.  Of  a  mill  worth  $78,000,  B.  owned  $2365,  C.  owned 
$3600,  and  A.  owned  the  remainder.     What  was  A/s 
share  worth  ? 

10.  I  paid  $65  for  a  harness,  $147  for  a  carriage,  and 
for  a  horse  as  much  as  for  the  carriage   and   harness. 
What  was  the  cost  of  all  ? 

11.  What  is  the  difference  between  a  million  and  a 
thousand  ? 

12.  What  is  the  difference  between  a  hundred  million 
and  a  hundred  thousand  ? 

13.  25  masons  earn  $13  each  and  20  laborers  $8  each  per 
week.     How  many  dollars  do  they  all  earn  in  12  weeks  ? 

14.  How  many  days  in  a  year  ?     If  a  man's  income  is 
$3000  a  year  and  his  daily  expenses  average  $7.68,  what 
does  he  save  in  a  year  ? 

15..  A  merchant  bought  a  piece  of  broadcloth  contain- 
ing 56  yards  for  $133,  and  sold  it  at  $3.00  a  yard.  How 
much  did  he  make. 

16.  A  gentleman  paid  $2500  for  a  driving  -team  and 


MULTIPLICATION  113 

carriage,  the  team   cost  $380  more  than  the  carriage. 
What  was  the  value  of  the  horses  ? 

17.  A  speculator  bought  140  acres  of  land  for  $7560 
and  sold  86  acres  of  it  at  $75  an  acre,  and  the  remainder 
at  cost.     How  much  did  he  make  ? 

18.  Find  the  cost  in  dollars  and  cents  of  208  pounds  of 
coffee  at  28  cents  per  pound. 

19.  An  orchard  has  16  rows  of  apple-trees,  and  each 
row  has  27  trees  in  it.     If  30  bushels  are  gathered  from 
each  tree,  how  many  bushels  will  the  orchard  produce  ? 

20.  Two  men  are  950  miles  apart;  they  travel  towards 
each  other,  one  at  the  rate  of  30  miles  per  day,  and  the 
other  at  the  rate  of  42  miles.     At  the  end   of  8  days, 
how  far  will  they  be  apart  ? 

21.  A  merchant's  profits  from  his  business  were  $8695. 
If  he  paid  $869  for  house  rent,  and  three  times  as  much 
for  other  expenses,  how  much  did  he  save  ? 

22.  A  drover  bought  106  oxen,  at  $35  a  head;  it  cost 
him  $6  a  head  to  get  them  to  market,  where  he  sold  them 
at  $47.     Did  he  gain  or  lose,  and  how  much  ? 

23.  A  flour  merchant  bought  936  barrels  of  flour  at  $9 
a  barrel.     He  sold  480  of  them  at  $9.50  a  barrel,  and  the 
remainder  at  $8.50.     What  was  the  profit  or  the  loss  to 
him? 

24.  Two. vessels  start  from  the  same  port,  and  travel  in 
opposite  directions — one  at  the  rate  of  75  miles  a  day, 
the  other  at  the  rate  of  85  miles  a  day.     How  far  apart 
will  they  be  at  the  end  of  1 5  days  ? 

25.  Find  the  value  of: 

1.  99  X  8  +  51  x  10  —  7  X  104  +  26. 

Suggestion  :  Multiply  before  adding  or  subtracting. 
8 


114  ELEMENTARY  ARITHMETIC 

2.  56  +  17  +  13  X  4  —  5  x  15  +  8  X  81. 

3.  3976  X  23  +  3456  X  25  —  2879  X  34. 

4.  $10.11  X  924  —  $10.10  X  777  +  $2.48  X  123. 


DIVISION. 

INDUCTIVE   STEPS. 

1.  Since  2  +  2  or  two  times  2  =  4,  how  many  2's  are 
there  in  4  ? 

2.  Since  3  +  3  +  3  or  three  times  3  =  9,  how  many 
3's  are  there  in  9  ? 

3.  How  many  4's  are  there  in  eight?     In  12?     In  20? 

4.  Then  4  is  contained  how  many  times  in  8?     In  12? 
In  20? 

5.  How  many  times  is  : 

1.  $3  contained  in  $6  ?    $4  contained  in  $12  ?    $5 

in  $20  ? 

2.  6  books  contained  in  18  books  ?    7  gallons  con- 

tained in  21  gallons  ? 

The  process  by  which  we  determine  that  7  is  contained 
in  21  three  times  is  called  Dividing-. 

6.  Divide: 

1.  10  quarts  by  2  quarts.     10  quarts  by  5  quarts. 

2.  18  horses  by  3  horses.     30  men  by  10  men. 

3.  45  pounds  by  9  pounds.     12  weeks  by  6  weeks. 

DEFINITIONS. 

1.  Division  is  the  process  of  finding  how  many  times 
one  number  is  contained  in  another,  or  of  finding  one  of 
the  equal  parts  into  which  a  number  is  to  be  separated. 

The  latter  part  of  this  definition  is  discussed  on  page  115,  under  the 
head  of  Equal  Parts. 


DIVISION  115 

2.  The  Dividend  is  the  number  to  be  divided. 

3.  The   Divisor   is    the   number   used   in   effecting   a 
division. 

4.  The  Quotient  is  the  result  of  a  division,  and  answers 
the  question,  How  many  times  ?  or,  What  is  one  of  the 
equal  parts  ? 

5.  The  Remainder  is  the  part  of  the  dividend  left  after 
an  incomplete  division. 

6.  The  Signs  of  division  are  -5-, ,  and  ).    Each  one 

is  read  "  divided  by." 

7.  The  dividend  is  placed :  On  the  left  of  -5-,  as  in  12 

_=-  4  =  3;  above  ,  as  in  ^  =  3;  on  the  right  of  ), 

as  in  4)12,  or  4)12(3. 

3  12 

0 

Each   of  these  equations  is   read,  "12  divided  by  4 
equals  3." 

Equal  Parts. 

1.  When  anything  is  divided  into  two  equal  parts,  is 
not  each  part  called  one- half  of  that  thing? 

2.  What  is  one-half  of  4  ?     Of  8  ?     Of  10  ?     Of  12  ? 

3.  Write  the  division  of  4,  8,  10,  and  12  by  2  in  each 
of  the  ways  explained  in  7. 

4.  When  anything  is  divided  into  3  equal  parts,  can 
you  tell  what  each  part  is  called  ? 

5.  What  is  one-third  of  6  ?     Of  9  ?     Of  12  ?     Of  15  ? 

6.  By  means  of  signs  express  the  division  of  6,  9,  12, 
and  15,  by  3. 

7.  We  see  that  the  divisor  gives  the  name  to  the  equal 
parts. 


116  ELEMENTARY  ARITHMETIC 

1.  Divisor    2   names    each    equal    part    one-half, 

written  l. 

2.  Divisor   3   names   each   equal    part    one-third, 

written  ^. 

3.  Divisor  4  names   each  equal  part   one-fourth, 

written  J. 

4.  Divisor   5    names    each    equal    part    one-fifth, 

written  \. 

5.  Divisor   9   names   each   equal    part    one-ninth, 

written  ^. 

8.  Dividends  denote  the  number  of  equal  parts,  as  J, 
f,  £,  read  "one-half,  two-thirds, /owr-fifths." 

9.  Equal  parts  thus  indicated  are  called  Fractions. 

ORAL    EXERCISES. 

1.  In  the  fraction  f ,  point  out  the  dividend  and  divisor. 

2.  In  the  fraction  ^,  point  out  dividend  and  divisor, 
and  name  the  quotient. 

3.  To  find  the  quotient,  what  part  of  12  did  you  take  ? 

4.  How,  then,  do  you  find  J  of  a  number  ?    One-ninth  ? 

One-twentieth  ? 

5.  Read  the  following  fractions  : 

*•  T>  Ti  TIT'  A?  ~S">  f?  TT>  IT?  1T>  TT'  T5">  "5T- 
*•   T§">  2T>  A'  TT3">  "3T>  A>  "5~0">  TW  "2TO">  AlT* 

"•  T^*  T^J  4T>  iW?  "9T?  rainr?  1260'  1728'  1 900  • 

6.  "What  is  the  quotient  of: 

1.     9 --3?  6.  16-^-4?  11.  28  -^-4? 

2.36-«-6?  7.  49 -f- 7?  12.28-7-7?' 

3.  63  -T-  7  ?  8.  45  -5-  5  ?  13.  27  --  9  ? 

4.  40--4?  9.56-^7?  14.54^-9? 
6.15-7-5?  10.21-7-3?  15.42-7-6? 


DIVISION 


7.  What  is  the  quotient  of: 

1.  y?          3.  ^?          5.  ff?          7. 

272?  A      1009  £     1 0.8  9  Q 

•       6     •  *•       10     •  "•       I  2 

8.  What  is  the  quotient  of:  « 


5)35? 

(2.) 
5)25? 

(3.) 

8)32? 

9)36? 

(5.) 
7)63? 

(6.) 

11)88? 

(10.) 
3)36? 

(7.) 
8)56? 

(11.) 
4)44? 

(8.) 
11)132? 

(12.) 
5)60? 

(9.) 

2)24? 

(13.) 

6)66? 

(14.) 
7)63? 

(15.) 

8)96? 

(16.) 

12)144? 

ORAL   PROBLEMS. 

1.  How  many  pounds  of  sugar,  at  5  cents  a  pound, 
can  be  bought  for  35  cents. 

SOLUTION. — Since  1  pound  costs  5  cents,  for  35  cents  as  many  pounds 
can  be  bought  as  5  cents  is  contained  times  in  35  cents,  or  7  pounds. 

When  the  quotient  answers  How  many  times,  are  not  the 
dividend  and  divisor  like  numbers  ? 

2.  If  a  man  earns  $3  per  day,  how  long  will  it  take 
him  to  earn  $12? 

3.  When  eggs  are  selling  for  12  cents  a  dozen,  how 
many  dozen  will  60  cents  buy  ? 

4.  For  21  cents  how  many  pens  can  be  bought  at  3 
cents  each  ? 

5.  How  many  times  can  5  gallons  be  taken  from  a 
cask  containing  30  gallons  ? 

6.  If  a  man  can  dig  7  rods  of  ditch  in  a  day,  how 
many  days  will  it  take  him  to  dig  28  rods  ? 


118  ELEMENTARY  ARITHMETIC 

7.  If  a  man  pays  $56  for  seven  yards  of  cloth,  how 
much  is  that  a  yard  ? 

SOLUTION.— Since  7  yards  cost  $56,  one  yard  costs  \  of  $56,  or  $8. 

By  what  did  you  divide  t*  take  -f  of  56  ? 

Did  you  divide  by  7  yards  or  by  the  abstract  number  7  ? 

8.  How  much  hay,  at  $8  per  ton,  can  be  bought  for  $32  ? 

9.  If  $55  is  paid  for  11  cords  of  wood,  what  is  the 
price  per  cord  ? 

10.  If  a  farm  of  120  acres  is  divided  into  12  equal  lots, 
how  many  acres  does  each  lot  contain  ? 

11.  There  are  96  trees  in  an  orchard,  and  12  trees  in 
each  row.     How  many  rows  are  there  ? 

12.  In  12  hours  108  horses  crossed  a  bridge.     What 
was  the  average  number  per  hour  ? 

13.  A  man  who  had  48  acres  of  land  divided  it  into  6 
fields  of  equal  size.     What  part  of  the  whole  was  each 
field  ?     How  many  acres  were  in  each  field  ? 

14.  If  I  deposit  $12  in  a  savings  bank  every  month,  in 
how  many  months  shall  I  deposit  $132  ? 

15.  If  6  car-loads  of  freight  weigh  18  thousand  pounds, 
how  much  does  each  car-load  weigh  ? 

16.  A  merchant  paid  144  cents  for  an  advertisement  of 
12  lines.     What  was  the  rate  per  line  ? 

17.  If  6  pairs  of  boots  cost  54  dollars,  what  will  10 
pairs  cost? 

ANALYSIS. — If  6  pairs  cost  54  dollars,  1  pair  costs  £  of  54  dollars,  or 
9  dollars,  and  10  pairs  cost  10  times  9  dollars,  or  90  dollars. 

18.  In  one  week  there  are  7  days.     How  many  weeks 
are  there  in  25  days,  and  how  many  days  remain  ? 

19.  Referring  to  the  previous  example,  7  times  3,  plus 
4,  equals  what  ? 


DIVISION  119 


PRINCIPLE. 

1.  When  the  dividend  and  divisor  are  like  num- 
bers, the  quotient  is  an  abstract  number  denoting 
"times." 

2.  When  the  divisor  is  an  abstract  number,  the 
dividend  and  quotient  are  like  numbers,  the  quo- 
tient being  a  part  of  the  dividend. 

3.  The  product  of  the  divisor  and  quotient,  plus 
the  remainder,  equals  the  dividend. 


The  Divisor  a  Single  Figure. 

WRITTEN  EXERCISES. 
1.  Divide  1728  by  4. 

Process.  Explanation. 

Divisor.  Dividend.  Quotient.  Beginning  at  the  left,  we  perceive  that  4 

4  )  1728  (  432  wm  not  divide  1,  but  1  thousand  +  7  hun- 

16  dreds  =  17  hundreds  ;   17  hundreds  -^-  by  4 

i  9  =  4  hundreds,  with  1  hundred  remaining ; 

.,  2  1  hundred  -f  2  tens  =  12  tens  ;   12  tens  -=- 

by  4  =  3  tens,  with  no  tens  remaining ;  the 

8  units  -i-  hy  4  =  2  units.     Hence,  the  quo- 

8  tient  is  432,  with  0  remaining. 

0  Proof. 

432  X  4  =  1728. 

The  preceding  process  is  called  Long  Division. 

A  second  process,  called  Short  Division,  is  as  follows : 

Process.  Explanation. 

4)172&  After  dividing  17  hy  4,  the  one   hundred   remaining, 

430  instead  of  being  written,  is  prefixed  mentally  to  the  2  tens, 

making  12  tens. 


120 


ELEMENTARY  ARITHMETIC 


2.  Solve  both  by  long  and  short  division  the  following : 
1.     963  -*-  3.     11.  2450  -^-  2.     21.  $49.86  +  9. 


2.  1926  —  6. 

3.  3672  —  4. 

4.  3246  —  6. 

5.  2763  —  9. 

6.  3235  —  5. 

7.  2205  —  7. 

8.  2696  —  8. 

9.  3456  —  9. 
10.  2465  —  5. 

3.  Divide  607  by  2. 

Process. 
2)607 

303  —  1  rem. 


12.  3252  —  3. 

13.  4872  —  4. 

14.  6830  —  5. 

15.  2976  —  6. 

16.  2985  —  5. 

17.  4635  —  3. 

18.  3986  —  4. 

19.  3248  —  8. 

20.  5256  —  6. 
Also,  617  by  2. 

Process. 
2)617(308 


22.  $67.65  —  5. 

23.  $38.36  —  7. 

24.  $98.72  —  8. 

25.  $45.60  —  5. 

26.  $83.34  —  6. 

27.  $81.72  —  9. 

28.  $91.84  —  4. 

29.  $26.64  —  3. 

30.  $27.37  —  7. 


17 


1 


Explanation. 

In  the  first  case  6  -=-  2  = 
3;  o-f-2  =  0;  7  -=-2  =  3, 
with  1  remaining. 

In  the  second  case  6  -j-  2 
—  3 ;  1  brought  down  and 
divided  by  2  =  0 ;  1  ten  and 
the  7  units  brought  down  = 


17  units ;  17  units  divided  by  2  =  8  units,  with  1  unit  remaining. 

Prove  by  the  principle. 

4.  Find  the  quotients  and  remainders  of: 


1.  737  - 

-2. 

10.  4055  - 

-6. 

2.  736  - 

-3. 

11.  9767  - 

-8. 

3.  963  - 

-4. 

12.  9896  - 

-5. 

4.  916  - 

-  5. 

13.  9894  - 

-9. 

5.  824  - 

-6. 

14.  9568  - 

-3. 

6.  937- 

-7. 

15.  9830  - 

-8. 

7.  995  - 

-8. 

16.  9847  - 

-7. 

8.  965- 

-9. 

17.  7408  - 

-7. 

9.  679  - 

-8. 

18.  3943  - 

-4. 

19.  48,743  ~  2. 

20.  53,742  —  3. 

21.  65,889  —  4. 

22.  47,397  —  5. 

23.  67,327  —  7. 

24.  49,538  —  6. 

25.  93,570  —  7. 

26.  96,674  —  8. 

27.  57,487  —  9. 


DIVISION 


121 


28.  986  —  5.         39.  4792  — 

29.  927  —  6.         40.  4765  — 

30.  965  —  7.         41.  4827  — 

31.  995  —  4.         42.  5672  — 

32.  503  —  7.         43.  7484  — 

33.  765  —  2.         44.  9832  — 

34.  998  —  8.         45.  9323  — 

35.  963  —  5.         46.  9524  — 

36.  965  —  9.         47.  6498  — 

37.  957  _  4.         48.  8110  — 

38.  974  —  6.         49.  1474  — 
With  the  divisor  and  remainder 

examples  make  a  fraction ;  thus  (1) 
Write  the  exact  quotient  of  all 
second  column;  thus  (21)  675f. 


9.  50.  38,596  —  3. 

9.  51.  98,696  —  5. 

4.  52.  96,575  —  7. 

6.  53.  94,701  —  8. 

6.  54.  69,886  —  9. 

7.  55.  -86,340  —  3. 
9.  56.  90,678  —  4. 

6.  57.  99,365  —  6. 
4.  58.  93,720  —  3. 

7.  59.  94,831  —  8. 
6.  60.  95,942  —  9. 
of  each  of  the  above 
|,  (2)  i,  (3)  |,  etc. 
the  examples  in  the 


"WRITTEN   PROBLEMS. 

1.  How  many  6-pound  packages  of  buckwheat  flour 
can  be  made  from  1200  pounds  of  flour? 

2.  How  many  ploughs  at  $7  each  can  be  bought  for 
$1232? 

3.  How  many  tons  of  coal  at  $7  a  ton  can  be  bought 
for  $1995? 

4.  There  are  3  feet  in  one  yard.     How  many  yards 
are  there  in  63,360  feet  ? 

5.  There  are  8  quarts  in  a  peck.     How  many  pecks 
in  525,232  quarts  ? 

6.  4  pecks  make  1  bushel.     How  many  bushels  do 
249,024  pecks  make  ? 

7.  If  9  cents  will  buy  one  pound  of  cotton,  how  many 
pounds  will  507,285  cents  buy? 


122  ELEMENTARY  ARITHMETIC 

8.  If  $4  will  buy  one  yard  of  velvet  and  $1  will  buy 
J  of  a  yard,  how  many  yards  and  fourths  of  a  yard  can 
be  bought  for  $23  ? 

9.  How  many  times  must  you  take  $7  to  make  $1134  ? 
How  many  times  must  you  take  $9  ? 

10.  A   stage    travelled   8   miles    per    hour.      In   how 
many   hours   would   it   travel,   at  the   same   rate,   4704 
miles  ? 

11.  A  bicycler  rides  at  the  rate  of  12  miles  per  hour. 
How  long  would  it  take  him  to  ride  1728  miles? 

12.  James  Blair  paid  $37,504  for  some  Western  land 
at  $8  per  acre.     How  many  acres  did  he  buy  ? 

13.  A  man  gave  $36,755  in  equal  shares  to  his  5  chil- 
dren.    How  much  did  he  give  to  each  child  ? 

14.  If  sound  moves  10,080  feet  in  9  seconds,  how  many 
feet  does  it  move  in  1  second  ? 

15.  9  square  feet  make  1   square   yard.     How  many 
square  yards  in  43,560  square  feet  ? 

16.  7  days  equal  1   week.      In   364  days   how  many 
weeks  ? 

17.  How  many  yards  of  cloth,  at  9  cents  per  yard,  can 
be  bought  for  58,878  cents  ? 

18.  How  many  $5  bills  must  be  counted  out  to  pay  a 
bill  of  $1240. 

19.  How  many  pounds  of  sugar,  at  8  cents  a  pound, 
must  be  given  for  488  pounds  of  coffee,  at  22  cents  a 
pound  ? 

20.  Find  the  value  of  240  bottles  of  wine  at  $3  a  bottle, 
and  find  how  many  $4  bottles  of  wine  must  be  given  in 
exchange  for  the  $3  bottles. 

21.  Prove  that  if  we  multiply  a  given  number  by  100 


DIVISION  123 

and  divide  the  product  by  4  we  obtain  the  same  result  as 
when  multiplying  by  25.  ***** 


***** 


22.  Prove  by  the  asterisks  in  the  margin 

*  *  *  *  * 

that  4X5  =  5x4. 


The  Divisor  a  Single  Digit  with  Ciphers  Annexed. 
ORAL,   EXERCISES. 

1.  If  50  cents  be  put  in  10  equal  packages,  how  many 
cents  will  each  package  contain  ? 

2.  When  flour  is  worth  $10  a  barrel,  how  many  barrels 
can  be  bought  for  $110? 

3.  120  divided  by  10  gives  what  quotient?     What 
remainder  ?     Does  not  cutting  off  the  cipher,  thus  12/0, 
give  you  the  same  results  ? 

4.  How  many  times  is  10  contained  in  124?     What 
remains  ?     Does  not  cutting  off  the  4  thus,  12/4,  give  the 
same  results  ?     What  is  the  exact  quotient  ? 

5.  How,  by  cutting  off,  can  you  divide  124  by  100? 
What  is  the  quotient  ?     What  is  the  remainder  ?     What 
is  the  exact  quotient  ? 

6.  How  many  figures  will  you  cut  off  to  divide  by 
1000? 


PRINCIPLES. 

1.  Cutting  off  one  figure  divides  by  1O. 

2.  Cutting  off  two  figures  divides  by  1OO. 

3.  Cutting  off  three  figures  divides  by  1OOO,  and 
so  on. 


124  ELEMENTAKY  ARITHMETIC 

WRITTEN   EXERCISES. 

1.  Divide  365  by  10. 

Process.  Explanation. 

Divisor.  Dividend.  In  accordance  with  Principle  1,  to  divide 

10  )  36[5  by  10,  we  cut  off  one  figure  from  the  divi- 

36^j-,  Quotient.  dend,  and  obtain  for  quotient  36,  with  5 

as  remainder.     Hence,  the  exact  quotient 

is  36^. 

2.  Divide  1728  by  300. 

Process.  Explanation. 

3|OQ)  17|28  In   accordance  with  Principle   2,   to  divide  by 

52 28  100  we  cut  off  two  figures,  and  obtain  for  quotient 

17,  with  28  remaining.     Since  the  entire  divisor  is 

300,  we  must  divide  this  quotient  by  three.  One-third  of  17  hundreds  = 
5  hundreds,  with  two  hundreds  remaining ;  2  hundreds  =  200  units  ;  200 
units  plus  28  units  =  228  units.  Hence,  the  exact  quotient  is  6f $$. 

Hence,  the  following  rule  : 

1.  Out  off  the  ciphers  from  the  divisor. 

2.  Cut  off  as  many  figures  from  the  dividend. 

3.  Divide  by  the  significant  figure  of  the  divisor. 

4.  Make  a  fraction  with  the  remainder  and  divisor. 

To  apply  the  rule : 
3.  Divide  1898  by  400. 

Process.  Explanation. 

4100  )  18198  Having  cut  off  the  two  ciphers  from  400  and  two 

/i  2  9  8 '  figures  from  the  dividend,  we  obtain  18  for  quotient 

and  98  for  remainder.  Dividing  now  by  the  signifi- 
cant figure  4,  we  obtain  4  for  quotient  and  200  -{-  98  for  remainder. 
Making  a  fraction  with  the  remainder  and  divisor,  we  have  for  the  exact 
quotient  4f$$. 


DIVISION  125 

4.  Divide : 

1.  89  by  10.  16.  96,704  by  3000. 

2.  376  by  100.  17.  54,970  by  4000. 

3.  3422  by  100.  18.  49,685  by  5000. 

4.  5489  by  1000.  19.  74,769  by  6000. 

5.  6079  by  1000.  20.  82,546  by  7000. 

6.  71,560  by  100.  21.  99,839  by  8000. 

7.  97,048  by  100.  22.  99,953  by  9000. 

8.  57,084  by  1000.  23.  123,456  by  1000. 

9.  97,068  by  1000.  24.  789,012  by  2000. 

10.  95,650  by  1000.  25.  345,678  by  3000. 

11.  7936  by  40.  26.  9,012,345  by  40,000. 

12.  3079  by  50.  27.  6,789,012  by  50,000. 

13.  4987  by  300.  28.  3,456,789  by  60,000. 

14.  5097  by  500.  29.  1,982,734  by  70,000. 

15.  90,798  by  2000.  30.  5,678,901  by  80,000. 

The  Divisor  Any  Number  of  Dig-its. 

EXERCISES. 
1.  Divide  25,003  by  48. 

Process.  Explanation. 

48  )  25003  (  520f|  Beginning  at  the  left,  we  perceive  that  48 

240  will  divide  neither  2  nor  25.     We  therefore 

JQQ  divide  48  into  250  hundreds,  and  obtain  5 

C)g  hundreds  for  quotient.     48  X   5  hundreds  = 

To  240  hundreds  ;  250  hundreds  —  240  hundreds 

=  10  hundreds.     Annexing  0  tens  from  the 

dividend,  we  have  100  tens ;  dividing  100  tens  by  48,  we  obtain  2  tens. 
48  X  2  tens  =  96  tens  ;  100  tens  —  96  tens  =  4  tens.  Annexing  3  units 
from  the  dividend,  we  have  43 ;  48  is  contained  in  43  no  times,  and  we 
write  0  in  the  quotient.  43  is  the  remainder,  and  the  exact  quotient  is 
620ft. 


126 


ELEMENTARY  ARITHMETIC 


Proof. 

520  X  48  +  43  =  25,003. 


Help  Table. 


48  X  0  =  0 
48  X  1  —  48 
48  X  2  =  96 
48  X  3  =  144 
48  X  4  =  192 


48  X  6  ==  240 
48  X  6  =  288 
48  X  7  =  336 
48  X  8  =  384 
48  X  9  =  432 


What  method  was  employed  above,  Long  Division  or 
Short  Division  ? 

Look  at  the  table  and  state  why  neither  4  nor  6  can  be 
taken  for  the  first  figure  of  the  quotient. 

BULB. 

1.  Begin  at  the  left  to  divide. 

2.  Divide  the  divisor  into  the  fewest  figures  that  will 
contain  it. 

3.  Multiply  the  divisor  by  the  quotient  figure. 

4.  Subtract,  and  annex  the  next  figure  of  the  dividend. 

5.  Divide  the  divisor  into  the  number  thus  formed. 

6.  Multiply,  subtract,  and  annex  the  next  figure  of  the 
dividend. 

7.  So  proceed  until  all  the  figures  of  the  dividend  have 
been  used. 

8.  Finally,  make  an  exact  quotient,  if  necessary. 

2.  Divide  304,675  by  48,  using  the  above  table. 

3.  Divide  789,101  by  56,  first  having  made  a  table  to 
aid  you  in  this  work. 

4.  Divide  123,456  by  27. 


DIVISION  127 

5.  Divide: 

1.  7890  by  10.  31.  15,780  by  15. 

2.  5303  by  11.  32.  15,909  by  16. 

3.  3667  by  12.  33.  14,668  by  17. 

4.  7590  by  13.  34.  37,950  by  18. 

5.  4089  by  14.  35.  24,534  by  19. 

6.  9085  by  15.  36.  63,595  by  20. 

7.  2476  by  16.  37.  17,332  by  30. 

8.  3420  by  17.  38.  27,360  by  50. 

9.  2599  by  18.  39.  20,792  by  60. 

10.  3858  by  19.  40.  34,652  by  70. 

11.  4420  by  20.  41.  44,200  by  80. 

12.  4605  by  21.  42.  46,050  by  90. 

13.  4807  by  23.  43.  14,421  by  100. 

14.  4602  by  34.  44.  13,806  by  101. 

15.  6095  by  45.  45.  24,380  by  102. 

16.  4555  by  56.  46.  22,775  by  121. 

17.  4478  by  67.  47.  26,868  by  123. 

18.  4135  by  78.  48.  28,945  by  130. 

19.  3708  by  89.  49.  29,664  by  144. 

20.  7334  by  99.  50.  66,006  by  156. 

21.  8178  by  73.  51.  81,780  by  160. 

22.  4952  by  93.  52.  54,472  by  172. 

23.  6840  by  82.  53.  82,080  by  185. 

24.  5198  by  71.  54.  51,980  by  190. 

25.  7716  by  60.  55.  17,716  by  200. 

26.  8840  by  40.  56.  17,680  by  205. 

27.  9210  by  42.  57.  27,630  by  221. 

28.  9614  by  46.  58.  38,456  by  276. 

29.  9204  by  68.  59.  46,020  by  290. 

30.  9110  by  57.  60.  54,660  by  223. 


128  ELEMENTARY  ARITHMETIC 

6.  Divide: 

1.  300,165  by  327.  17.  2,010,081  by  7292. 

2.  586,635  by  404.  18.  3,453,901  by  6007. 

3.  236,823  by  526.  19.  2,980,506  by  7035. 

4.  304,727  by  783.  20.  4,007,205  by  6294. 

5.  343,926  by  250.  21.  1,795,791  by  7272. 

6.  316,743  by  684.  22.  9,500,367  by  5959. 

7.  202,729  by  672.  23.  7,989,368  by  7979. 

8.  349,052  by  879.  24.  1,509,307  by  5757. 

9.  594,924  by  709.  25.  8,640,073  by  8383. 

10.  458,015  by  532.  26.  4,560,079  by  4949. 

11.  320,479  by  373.  27.  9,200,537  by  8787. 

12.  329,500  by  402.  28.  6,975,841  by  9696. 

13.  660,209  by  723.  29.  7,935,836  by  8484. 

14.  2,658,360  by  3245.  30.  5,009,703  by  7373. 

15.  3,360,843  by  5023.  31.  9,071,564  by  6969. 

16.  4,300,436  by  6436.  32.  3,795,411  by  8383. 

ORAL  PROBLEMS. 

1.  Hats   are  $4  apiece;    broadcloth  is  $6   per  yard. 
How  many  of  the  hats  will  pay  for  6  yards  of  the  broad- 
cloth ? 

2.  Twelve  lambs   cost  $24.     To  gain  $12,  how  must 
they  be  sold  per  head  ? 

3.  There  are  3  feet  in  a  yard.     How  many  yards  in 
627  feet? 

4.  A  boy  earns  $3.00  per  week.     After  paying  $1.00 
per  week  for  other  things,  how  long  will  it  take  him  to 
earn  enough  to  buy  a  suit  of  clothes  costing  $16.00  ? 

5.  A  man  had  $60 ;  he  spent  one-half  of  it  for  sheep 
at  $3.00  each.     How  many  sheep  did  he  buy  ? 


DIVISION  129 

6.  A  train  goes  40  miles  per  hour.     How  long  will  it 
take  to  go  480  miles  ? 

7.  A  newsboy  bought  10  papers  for  35  cents  and  sold 
them  so  as  to  gain  15  cents.      How  much  did  he  get 
apiece  for  them  ? 

8.  I  bought  10  tons  of  coal  for  $50  and  sold  it  so  as  to 
gain  $20.     How  much  did  I  gain  per  ton  ? 

9.  A  lady  having  $5  bought  5  chickens  at  50  cents 
each  and  a  turkey  for  $1.50.     How  much  money  had  she 
left? 

10.  What  would  be  the  cost  of  1  arithmetic  if  $162.96 
were  paid  for  400  copies  ? 

11.  If  a  farmer  exchanges  6  firkins  of  butter,  worth 
$20  a  firkin,  for  cloth  at  $4  a  yard,  how  many  yards  will 
he  receive  ? 

12.  At  $5  per  gallon,  how  much  alcohol  can  be  bought 
for  $37. 

WRITTEN   PROBLEMS. 

1.  A  dividend  is  14,145,  and  the  quotient  123.    "What 
is  the  divisor  ?     State  the  principle. 

2.  A  certain  number  is  contained  41  times  in  1043, 
with  18  as  a  remainder.     What  is  the  number? 

3.  Paid  $3780  for  28  acres  of  land.     How  much  was 
that  per  acre  ? 

4.  How  many  bushels  of  corn  at  56  cents  each  can  be 
bought  for  7560  cents. 

5.  A  man  raised  6427  bushels  of  oats  upon  107  acres. 
What  was  the  average  crop  per  acre  ? 

6.  If  a  farmer  can  lay  up  $425  a  year,  how  many 
years  will  it  take  him  to  lay  up  $6800  ? 


130  ELEMENTARY  ARITHMETIC 

7.  A  field  of  600  acres  produced  8400  bags  of  wheat. 
How  many  bags  to  the  acre  was  that  ? 

8.  A  man  paid  $170,352  for  36  city  lots.     What  was 
the  average  cost  per  lot  ? 

9.  In  94,185  yards  of  sheeting  are  how  many  pieces, 
each  piece  containing  45  yards  ? 

10.  A  surveyor  travels  41,600  rods  in  one  week.    There 
being  320  rods  in  a  mile,  how  many  miles  does  he  travel? 

11.  The  distance  of  the  earth  from  the  sun  is  91,500,- 
000  miles.     Light  travels  at  the  rate  of  185,000  miles  per 
second.    In=how  many  seconds  does  light  travel  from  the 
sun  to  the  earth  ? 

12.  At  the  rate  of  $75  per  acre,  how  many  acres  of 
land  can  be  bought  for  40,425  dollars  ? 

13.  A  man  bought  133  horses  and  160  mules,  paying 
39,555   dollars  for  the  whole.     What  was  the  average 
price  ? 

14.  The  divisor  is  88,  the  quotient  $248,  and  the  re- 
mainder $66.     What  is  the  dividend  ? 

15.  The  product  of  two  numbers  is  40,796.     One  of 
the  numbers  is  124.     What  is  the  other  number  ? 

16.  How  many  times  must  a  pole  11  feet  long  be  ap- 
plied to  measure  a  distance  of  a  mile  (5280  feet)  ? 

17.  A  long  ton  of  coal  is  2240  pounds  and  a  short  ton 
is  2000  pounds.     Find  how  many  short  tons  weigh  as 
much  as  250  long  tons. 

18.  A  man  has  8000  dollars.     He  buys  two  houses  for 
4500  dollars,  and  some  land  at  140  dollars  per  acre.    How 
many  acres  will  he  get  ? 

19.  Bought  288  barrels  of  flour  at  $1482,  and  sold  the 
same  for  $2058.     What  was  the  gain  on  each  barrel  ? 


DIVISION  131 

20.  How  many  horses  at  $82.50  each,  can  be  bought 
for  $6187.50. 

21.  There  are  7000  Troy  grains  in  a  pound  avoirdu- 
pois.    How  many  pounds  are  there  in  118,125  grains. 

22.  3600  seconds  make  an  hour.     How  many  hours  in 
12,950  seconds. 

23.  How  many  schooners,  each  carrying  8700  bushels 
of  wheat,  will  be  required  to  carry  843,900  bushels  ? 

24.  How  long  can  125  men  subsist  on  an  amount  of 
food  that  will  last  1  man  4500  days  ? 

Parenthesis  and  Vinculum. 

1.  A  Parenthesis,  ( ),  signifies  that  the  numbers  within 
it  are  to  be  subjected  to  the  same  operation,  and  in  con- 
sequence the  operations  indicated  within  it  should  first  be 
performed. 

In  the  expression,  (16  —  4)  -s-  4,  we  first  subtract  4  from  16,  and  then 
divide  the  result  by  4. 

2.  A  Vinculum,  ,  may  be  used  instead  of  the 

parenthesis. 


6  —  4x4  and  (6  —  4)  X  4  have  precisely  the  same  meaning. 

EXERCISES. 
Find  the  value  of: 

1.  (423  +  47)  —  (492  —  326)  —  76. 

2.  (325  —  92)  —  (226  —  29  +  7)  -f  20. 

3.  (413  __  200)  —  (118  —  24^7  +  6)  +  3. 

4.  (282  —  97)  —  (4~xT+  38)  +  20. 

5.  (4  X  5  X  9)  —  (3  +  9)  -*-  4  +  6. 


6.  (6  +  2  +  7)  X  5  —  (8  +  9  +  4)  -*-  7  +  20. 

7.  (23  +  8  —  1)  X  6. 


132  ELEMENTARY  ARITHMETIC 

8.  24  —  7  +  18  X  7. 

9.  (22  —  3  +  26)  X  9. 

10.  (3  +  4)  X  9  —  (3  +  6)  -*-  3. 


11.  (6  +  8  —  4)  X  4  +  (203  +  67)  -f-  3. 


12.  (100  —  l)-r-9— (97  +  20)-r-13-f(4  -f  8 -s- 4. 

13.  (56  +  4)  -s-  6  +  (21  +  24)  -*-  (8  —  3)  +  7. 

14.  (105  -7-  21)  +  (80  -*-  5)  X  (81  +  36  —  9). 

15.  327  X  6  -v-  109  +  52  X  5  —  (42  +  8  X  4).  ' 

ANALYSIS. 

Analysis  reasons  from  the  given  number  to  one,  and 
tlien  from  one  to  the  required  number. 

PROBLEMS. 

1.  If  7  barrels  of  flour  cost  35  dollars,  what  will  9 
barrels  cost? 

Analysis.  Explanation. 

7  barrels  =  35  dollars.  If  7  barrels  cost  35  dollars,  1  barrel 

1  barrel   =    5  dollars.         costs  *  of  35  dollars'  or  5  dollars ;  if  l 

barrel  costs  5  dollars,  9  barrels  cost  9 

9  barrels  =  45  dollars.         times  5  dollarg)  or  45  dol]ars 

2.  If  7  men  can  do  a  piece  of  work  in  8  days,  in  how 
many  days  can  8  men  do  it  ? 

Analysis.  Explanation. 

7  men  —     8  days.  It  w^^  ta^e  1  man  longer  than  7  men  to 

1  man  =  56  days.         do  a  piece  of  work '  hence>  we  say  " If  7  men 

can  do  the  work  in  8  days,  1  man  will  do  it  in 

7  days.         ?  timeg  g  daygj  or  56  dayg .  and  .f  1  man  can 

do  it  in  56  days,  8  men  can  do  it  in  £  of  56  days,  or  7  days." 

3.  A  man  bought  sheep  at  the  rate  of  3  for  $18.     How 
many  did  he  buy  for  $906  ? 


DIVISION  133 

Analysis.  Explanation. 

3  sheep  =  $18.  If  3  sheep  cost  ^18)  1  sheep  costs 

.      '  i  of  $18,  or  $6  ;  if  1  sheep  costs  $6, 

for  $906  he  bought  as  many  sheep  as 

$906  -+-  $6  =  151  Sheep.  $6  is  contained  times  in  $906,  or  151 

sheep. 

4.  In  17  miles  there  are  29,920  yards.     How  many 
yards  are  there  in  35  miles  ? 

5.  If  12  barrels  of  flour  are  worth  $120,  what  will  36 
barrels  cost? 

6.  If  45  acres  produce  2520  bushels  of  corn,  how  many 
bushels  do  28  acres  produce  at  the  same  rate  ? 

7.  A  merchant  buys   168  barrels   of  flour  for  $672. 
How  many  barrels  can  he  buy  for  $984  at  the  same  rate  ? 

8.  If  a  man  can  travel  154  miles  in  7  days,  how  far 
can  he  travel  in  43  days  ? 

9.  If  a  stack  of  oats  will  serve  25  horses  9  days,  how 
many  days  will  it  serve  15  horses? 

10.  If  8  men  can  do  a  piece  of  work  in  145  days,  how 
long  will  it  take  29  men  to  do  the  same  work  ? 

11.  If  46  acres  of  land  produce  2484  bushels  of  corn, 
how  many  bushels  will  120  acres  produce  ? 

12.  At  6  cents  each,  how  many  oranges  can  be  bought 
for  $5.58  ? 

13.  If  29  tons  of  coal  cost  $116,  what  will  37  tons  cost? 

14.  What  is  the  cost  of  68  barrels  of  molasses  if  37 
barrels  cost  $888? 

15.  A  farmer  sold  18  calves  at  the  rate  of  3  for  $33. 
How  much  did  he  get  for  them  ? 

16.  If  19  horses  can  be  bought  for  1520  dollars,  how 
many  horses  can  be  bought  for  1360  dollars? 


134  ELEMENTARY  ARITHMETIC 

17.  How  long  will  it  take  19  men  to  do  a  piece  of  work 
that  17  men  can  do  in  133  days? 

18.  25  barrels  of  flour  weigh  4900  pounds.     Find  the 
weight  of  87  barrels. 

19.  If  36  men  can  be  hired  for  $50.40  for  one  day,  how 
many  men  can  be  hired  for  the  same  time  for  $133  ? 

20.  If  I  pay  $96  for  25  hats,  how  much  must  I  pay  for 
63  hats  at  the  same  rate  ? 

21.  A  certain  quantity  of  barley  lasts  11  horses  15  days. 
How  long  would  it  last  5  horses? 

22.  I  bought  9  horses  for  $1530,  and  sold  them  for 
$1665.     How  much  did  I  gain  on  each  horse? 

23.  If  13  hogsheads  of  molasses  can  be  bought  for  273 
dollars,  how   many  hogsheads   can  be  bought   for   609 
dollars  ? 

24.  A  man  bought  150  calves  at  $14  a  head  and  sold 
them  so  as  to  gain  $450.     What  did  he  get  a  head  ? 

25.  In  how  many  weeks  can  a  father  and  son  together 
earn  $71.75,  if  the  father  earns  $10.60  and  the  son  $3.75 
per  week  ? 

26.  What  cost  7  pieces  of  muslin,  37  yards  each,  at 
$.13  per  yard? 

27.  Engaged  in   farming,  I   wish    to    obtain  $370;    I 
therefore  sell  100  bushels  of  wheat  at  $0.75  per  bushel 
and  enough  apples  at  $2.50  per  barrel  to  obtain  the  sum 
required.     How  many  barrels  of  apples  do  I  sell  ? 

28.  A  man  bought  320  barrels  of  apples  at  $3  per  bar- 
rel.    He   found   120   barrels   worthless.     At  what   rate 
must   he   sell   the  rest  to  get  back   all  the  money  ex- 
pended ? 

29.  If  a  merchant  buys  coal  at  the  rate  of  $3.75,  and 


DIVISION  135 

sells  it  at  $5.00  per  ton,  how  many  tons  must  he  sell  in 
order  to  gain  $1500  ? 

30.  A  grocer  bought  250  pounds  of  coffee  for  $82.50, 
and  sold  it  at  37  cents  a  pound.     What  was  his  gain  ? 

31.  A  year  contains  365  days.     The  yearly  salary  of 
the   President   of  the  United  States  is    50,000   dollars. 
How  much  can  he  spend  daily  and  save  at  the  end  of  the 
year  9850  dollars  ? 

32.  If  16  horses  cost  $1952,  what  will  22  horses  cost  at 
$6  less  a  head  ? 

33.  If  6  dollars'  worth  of  flour  lasts  a  family  of  8  per- 
sons 2  months,  how  many  dollars'  worth  will  last  13 
persons  the  same  length  of  time  ? 

Suggestion :  8  persons  —  6.00  dollars  ;  1  person  =  .75  dollars  ;   13  per- 
sons =  how  many  dollars  ? 

34.  If  a  man  could  reap  a  field  of  oats  in  16  days  by 
working   12  hours  a  day,  in  how  many  days  could  he 
reap  it  by  working  8  hours  per  day  ? 

35.  I  sold  land  at  $46  an  acre,   and  received  for  it 
$4508.     How  much  would  I  have  received  for  it,  had  I 
sold  it  at  $58  an  acre  ? 

36.  A  man  bought  96  apples  at  the  rate  of  4  for  6 
cents,  and  exchanged  them  for  pears  at  8  cents  apiece. 
How  many  pears  did  he  receive  ? 

37.  Mr.    Moore,   having   $10,000,   invests  $3750  in   a 
house,  and  the  remainder  in  land  at  $125  an  acre.     How 
much  land  does  he  buy  ? 

38.  Find  the  value  of  (152  +  119  —  118)  -*-  9. 

39.  Find  the  value  of  (96  —  84  -=-  7)  X  10. 

40.  The  quotient  is  367,  the  divisor  is  445,  and  the  re- 
mainder 189.     What  is  the  dividend  ? 


136  ELEMENTARY  ARITHMETIC 

41.  In  payment  of  a  bill,  a  lady  gave  30  twenty-dollar 
bills,  10  ten-dollar  bills,  4  one-dollar  bills,  and  16  five- 
dollar  bills.     How  large  was  tbe  bill  sbe  paid  ? 

42.  A  pupil  being  told  to  multiply  a  certain  number 
by  23,  mistook  tbe  3  for  a  5,  and  his  answer  was  150. 
What  was  the  correct  answer  ? 

43.  If  20  men  can  do  a  piece  of  work  in  31  days,  in 
how  many  days  will  the  work  be  done,  if  11  more  men 
are  employed  ? 

44.  Find  the  value  of  (XL  V.  +  III.)  -f-  VI.  +  (X.  +  XV.) 
+  (VII.  —  II.)  +  VI. 

REVIEW. 

1.  Define  the  following  terms  : 

1.  Division.  6.  One-half. 

2.  Dividend.  7.  One-third. . 

3.  Divisor.  8.  Parenthesis. 

4.  Quotient.  9.  Vinculum. 

5.  Remainder.  10.  Analysis. 

2.  Repeat  from  memory  the  first  three  principles  of 
Division. 

3.  Repeat  the  next  three  principles  and  the  rule,  in 
four  parts,  derived  therefrom. 

4.  Repeat  the  eight  parts  of  the  rule  of  long  division. 

5.  Describe  the  signs  of  division. 

6.  Define,  also,  Multiplication,  Multiplicand,  Multiplier, 
Product,  Abstract  Number,  Denominate  Number. 

7.  Give  the  six  principles  of  multiplication. 

8.  Define  Subtraction,  Minuend,  Subtrahend,  Remainder. 

9.  Repeat  the  principles  and  the  rule  of  subtraction. 
1O.  Repeat  the  principles  and  the  rule  of  addition. 


FACTORING  137 

11.  Define: 

1.  Unit.  6.  Analysis. 

2.  Number.  7.  Decimal  Point. 

3.  Arithmetic.  8.  Arabic  System. 

4.  Notation.  9.  Roman  System. 

5.  Numeration.  10.  Zero. 

12.  Repeat   the   principles    of  both   the   Arabic   and 
Roman  systems. 


FACTORING. 

INDUCTIVE    STEPS. 

1.  2  X  5  =  10,  being  an  expression  of  equality,  is 
called   what?      2   and   5,   as   makers   of   10,   are   called 
what? 

2.  Make  an  equation  showing  that  2  and  3  are  factors 
of  6. 

3.  Will  each  of  the  factors  2  and  3  exactly  divide  6  ? 

4.  Make  an  equation   showing  the  product  of  two 
factors  equal  to  12. 

5.  If  your  equation  is  3  X  4  =  12,  will  3  and  4 
exactly  divide  12  ?     Is  not  a  factor,  then,  an  exact  divisor  ? 

6.  Has  3  any  factors  except  itself  and  1  ?     Has  4  any 
factors  except  itself  and  1  ? 

7.  Now  make  an  equation,  putting  3  factors  equal  to 
12,  and  excluding  1  as  a  factor? 

8.  In  the  following  numbers  point  out  each  that  has 
no  factors  except  itself  and  1 :  1,  £,  5,  4,  5,  #,  7,  8,  9. 

9.  Point   out   all   the   factors   of  these:    4,  6,  8,  9. 


138  ELEMENTARY  ARITHMETIC 

Which  one  has  the  greatest  number  of  factors  ?     What 
two  factors  will  produce  8  ? 

1O.  What  are  the  exact  divisors  of  the  following  num- 
bers :  16, 18,  20,  21,  43,  27,  29,  35,  42,  63,  17,  72,  37, 144. 

It  is  to  be  specially  noted  that  while  1  may  be  considered  a  factor  of  any 
number,  there  are  certain  numbers  that  can  have  no  factors  if  1  is  excluded. 
It  is  also  to  be  specially  noted  that  by  factors  and  divisors  are  meant  only 
integral  factors  and  divisors. 

DEFINITIONS. 

1.  An  Integral  Number  expresses  whole  units. 

23  and  16  are  integral  numbers,  or  integers. 

2.  An  Exact   Divisor   is   an  integer   that  divides  a 
number  without  a  remainder. 

2,  4,  6,  9  are  exact  divisors  of  36. 

3.  The  Factors  of  a  number  are  integers  that,  multi- 
plied together,  produce  the  number. 

7  and  8  are  factors  of  56. 

4.  A  Prime   Number   has  no  exact  divisors   except 
itself  and  1. 

1,  3,  5,  7,  11,  13  are  prime  numbers. 

5.  Prime  Factors  are  prime  numbers  used  as  factors. 

7  and  11  are  prime  factors  of  77. 

6.  A  Composite  Number  has  other  factors  than  itself 
and  1. 

8,  9,  12,  16  are  composite  numbers. 

7.  An  Even  Number  is  exactly  divisible  by  2. 

4,  10,  14,  22  are  even  numbers. 

8.  An  Odd  Number  is  not  exactly  divisible  by  2. 

1,  3,  5,  7,  9,  11  are  odd  numbers. 


FACTORING  139 

9.  Factoring-  is  the  process  of  finding  the  factors  of 
a  number. 

1O.  An  Exponent  is  a  figure  employed  to  show  how 
often  a  factor  occurs  in  a  given  number. 

8  =  2  X  2  X  2  =  23.     3  is  the  exponent. 


PRINCIPLES. 

1.  A  factor  of  a  number  is  an  integer. 

2.  A  factor  of  a  number  is  an  exact  divisor  of  it. 

3.  Only  a  prime  factor  of  a  number,  or  the  pro- 
duct of  two  or  more  of  its  prime  factors,  is  an 
exact  divisor  of  that  number. 


"WRITTEN   EXERCISES. 
1.  Find  the  prime  factors  of  468. 


Explanation. 

Since  the  prime  factors  of  468  are  exact  divisors  of  it, 
we  may  find  its  prime  factors  by  finding  the  prime  num- 
bers that  will  divide  it.  Dividing,  we  find  the  prime 
factors  of  468  to  be  2,  2,  3,  3,  13,  or  22,  32,  13. 

Proof. 
22  X  32  X  13  =  468. 


RULE. 

Using  continuously  the  least  prime  number  that  will 
serve  as  a  divisor,  divide  the  given  number,  and  the  suc- 
ceeding quotients  until  the  quotient  is  a  prime  number. 
The  divisors  and  the  last  quotient  will  be  the  prime  factors. 


140 


ELEMENTARY  ARITHMETIC 


2.  Find  the  prime  factors  of; 


1.  228 

12.  216 

23.  320 

34.  1250 

2.  324 

13.  484 

24.  500 

35.  1024 

3.  224 

14.  576 

25.  965 

36.  1728 

4.  344 

15.  432 

26.  719 

37.  1280 

5.  144 

16.  672 

27.  913 

38.  1152 

6.  225 

17.  396 

28.  745 

39.  3204 

7.  796 

18.  625 

29.  840 

40.  24024 

8.  576 

19.  912 

30.  990 

41.  4800 

9.  256 

20.  832 

31.  1008 

42.  6902 

10.  300 

21.  864 

32.  1120 

43.  8364 

11.  198 

22.  945 

33.  1176 

44.  10917 

CANCELLATION. 

INDUCTIVE    STEPS. 

1.  What  is  the  quotient  of  12  -f-  6  ? 

2.  What  are  the  factors  of  12? 

3.  What  are  the  factors  of  6  ? 

4.  In  your  thought,  put  the  factors  of  12  above  a  line. 

5.  Put  the  factors  of  6  below  the  line. 

6.  Reject  the  equal  factors  from  above  and  below  the 
line. 

7.  What  factor  remains  ? 

8.  Is  not  2  the  quotient  of  12  -v-  6. 

9.  Will  not  your  mental  picture  be  like  this :     ^\/ 3    ? 

1O.  Rejecting   equal   factors   thus  from  dividend   and 
divisor  is  called  Cancellation. 


CANCELLATION  141 


PRINCIPLE. 

Cancelling   the   same   factor  or  factors    in  both 
dividend  and  divisor  does  not  alter  the  quotient. 


WRITTEN   EXERCISES. 

1.  Divide  4x5x8Xl8by2x3X8Xl5. 

Process.  Explanation. 

2  Cancelling  2  and  4  we  have  a 

fy  nt 

A  NX  si  ^  a.  xx  r«  resulting  factor,  2  ;  cancelling  5 

?  X  TP  X  p  X  t-p  _  9  v  9  _  J.  j  i  r  i  •        * 

«       „       s       Tg  —  ^  A  ^  —  *r.  and  15  we  have  a  resulting  fac- 

f*    /N     P    X    P    X    A? 

3  tor,  3;   cancelling  3  and  18  we 
have  a  resulting  factor,  6  ;  can- 

celling resulting  factors  3  and  6  we  have  resulting  factor,  2  ;  cancelling 
8  and  8  we  have  cancelled  all  the  factors  of  the  divisor,  and  the  uncan- 
celled  factors  of  the  dividend,  2  and  2,  give  us  4  for  the  quotient. 

2.  Divide  4  X  2  X  8  X  24  by  36  X  8  X  2. 

Process.  Explanation. 

8  We  cancel  as  follows:  4  and  36, 

o__92  2  and  2,  8  and  8,  9  and  24.    8  remains 

-  ^" 


tf  V  %  "V  2        -  3  -      "T* 

A  P  A  />  ^  dividend,  3  as  divisor.     Hence,  the 

3  quotient  is  2|. 


RULES. 

1.  Cancel  out  of  both  dividend  and  divisor  all  common 
factors. 

2.  Divide  the  product  of  the  remaining  factors  of  the 
dividend  by  the  product  of  the  remaining  factors  of  the 
divisor. 

Query.  —  When  no  factor  remains  uncancelled,  what  is 
the  quotient  ? 


142  ELEMENTARY  ARITHMETIC 

3.  Divide,  using  cancellation : 

1.  48  X  3  X  4  by  8  X  7  X  5. 

2.  40  X  10  X  3  by  5  X  6  X  2. 

3.  4X5X7X9  by  2X2X6X7X3. 

4.  8  x  9  X  12  X  16  by  4  X  3  X  5  X  6  X  20. 

5.  2  X  3  X  8  X  12  X  24  by  6  X  4  X  36  X  4. 

6.  18  X  24  X  32  x  36  by  9  X  48  X  4  X  18. 

7.  40  X  18  X  13  X  8  by  10  X  13  X  16. 

8.  376  X  14  X  21  by  7  X  7  X  16  X  3. 

9.  5  X  25  X  874  by  2  X  437  X  5  X  5  X  5. 

10.  108  X  17  X  9  X  4  by  27  X  3  X  16  X  17. 

11.  15  X  4  X  8  X  9  by  30  X  2  X  6  X  12. 

12.  40  X  27  X  32  X  21  by  24  x  18  X  16  X  14. 

13.  30  X  36  X  24  X  42  by  45  X  27  X  8  X  28. 

14.  27  X  32  X  45  X  36  by  18  x  24  X  9  X  6. 

15.  6  X  7  X  9  X  11  by  2  X  3  X  7  X  3  X  21. 

16.  4  X  14  X  16  X  24  by  7  X  8  X  32  X  12. 

17.  11  X  9  X  7  X  15  X  6  by  30  X  3  X  21  X  3. 

18.  55  X  36  X  27  X  42  by  12  X  25  X  35  X  33. 

19.  36  X  64  X  25  X  40  by  32  X  50  X  18  X  10. 

20.  56  X  18  X  32  X  49  by  16  X  36  X  42  X  28. 

21.  11  X  39  X  14  X  96  by  44  X  18  X  26  X  14. 

22.  2  X  4  x  8  X  13  X  7  X  16  by  26  X  14  X  8. 

23.  125  X  60  X  24  X  42  by  25  X  120  X  36  X  5. 

24.  28  X  56  X  400  by  112  X  280. 

25.  56  X  36  X  35  X  24  by  40  X  48  X  21  x  18. 

WRITTEN   PROBLEMS. 

1.  How  many  dozen  eggs,  worth  15  cents  a  dozen, 
must  be  given  for  20  pounds  of  sugar  worth  5  cents  a 
pound  ? 


CANCELLATION  143 


Process.  Explanation. 


_  2Q.  _  £2  The  value  °f  the  SU»ar  =  2°   X    5  ' 

1  dozen  eggs  =  15  cents;  hence,  for  20  X  5 

cents  as  many  dozen  eggs  must  be  given  as  15 
is  contained  times  in  20  X  5,  which  by  cancellation  we  find  to  be  6|. 

2.  How  many  books,  at  15  cents  apiece,  may  be  ex- 
changed for  12  reams  of  paper  at  55  cents  a  ream  ? 

3.  How  many  pounds  of  butter,  at  24  cents  a  pound, 
will  pay  for  45  yards  of  dress  goods  at  32  cents  a  yard  ? 

4.  How  many  pounds  of  butter,  worth  15  cents  a  pound, 
may  be  bought  for  25  pounds  of  tea  at  48  cents  a  pound  ? 

5.  A  boy  bought  5  hens  at  20  cents  each  and  paid  for 
them  with  apples  at  10  cents  a  dozen.     How  many  dozen 
were  required  ? 

6.  How  many  tons  of  hay,  at  $16  per  ton,  must  be 
given  for  12  barrels  of  flour  at  $6  per  barrel? 

7.  If  a  farmer  sells  25  bushels  of  wheat  at  60  cents  a 
bushel  and  takes  his  pay  in  cloth  at  40  cents  a  yard,  how 
many  yards  of  cloth  does  he  get  ? 

8.  How  many  dozen  eggs,  at  28  cents  a  dozen,  will 
pay  for  84  pounds  of  sugar  at  5  cents  a  pound  ? 

9.  How  many  bushels  of  potatoes,  at  75  cents  a  bushel, 
must  a  farmer  give  for  36  yards  of  carpet  worth  $1.25  a 
yard? 

10.  How  many  bushels  of  oats,  at  40  cents  a  bushel, 
must  be  given  for  1600  bushels  of  wheat  at  75  cents  a 
bushel  ? 

11.  How  many  pieces  of  cotton  cloth,  each  piece  con- 
taining 42  yards,  at  8  cents  per  yard,  can  be  bought  for 
12  firkins  of  butter,  each  containing  56   pounds,  at  20 
cents  a  pound  ? 


144  ELEMENTARY  ARITHMETIC 

12.  If  18  men  can  do  a  piece  of  work  in  42  days,  how 
long  will  it  take  21  men  to  do  the  same  work  ? 
2 

^fCt     vx     -|  O 

Suggestion  :  ^  *  —  .     It  will  require  1  man  18  times  42  days. 


13.  If  1920  bricks  will  build  a  wall  15  yards  long,  how 
many  bricks  will  be  required  for  a  similar  wall  24  yards 
long? 

14.  By  travelling  at  the  rate  of  20  miles  a  day,  a  person 
can  complete  a  journey  in  18«days.     At  what  rate  must 
he  travel  to  finish  it  in  15  days  ? 

15.  How  many  pounds  of  coifee  at  24  cents  a  pound 
would  be  required  to  pay  for  3  hogsheads  of  sugar,  each 
weighing  1464  pounds,  and  worth  4.5  cents  a  pound  ? 

16.  A  gardener  sells  75  crates  of  berries,  24  boxes  in  a 
crate,  at  8  cents  a  box,  and  receives  in  return  12  rolls  of 
matting,  40  yards  in  a  roll.     Find  the  price  of  the  mat- 
ting per  yard  ? 

17.  Divide  285,120  by  5184,  using  the  prime  factors  of 
each  and  cancelling. 

NOTE.  —  Solve  by  cancellation  the  problems  under  the  head  of  Analysis, 
pages  132  to  136. 


FRACTIONS. 

INDUCTIVE    STEPS. 

1.  On  page  115  we  learned  that  a  Fraction  expresses 
one  or  more  of  the  equal  parts  of  anything.  One  of  two 
equal  parts  is  denoted  by  -J-,  read  "  one-half."  One  of 
three  equal  parts  is  denoted  by  ^,  read  "  one-third." 
One  of  six  equal  parts  is  denoted  by  £,  read  "  one-sixth." 


FRACTIONS  145 

Two  of  six  equal  parts  is  denoted  by  f ,  read  "  two-sixths." 
Six  of  six  equal  parts  is  denoted  by  f,  read  "six- 
sixths.'  " 

2.  If  an  apple  is  divided  into  six  equal  parts,  f  equals 
how  much  of  the  apple  ? 

3.  If  an  apple  is  divided  into  10  equal  parts,  what 
fraction  will  express  the  whole  of  the  apple  ? 

4.  If  an  apple  is  divided  into  20  equal  parts,  what  is 
one  part  called  ? 

5.  What  fraction  will  denote  the  whole  of  the  apple  ? 

6.  If  100  cents  are  divided  equally  among  5  boys,  what 
part  of  the  money  will  each  boy  receive  ? 

7.  What  is  £  of  100  cents  ? 

8.  Twenty  cents  is  what  part  of  100  cents  ? 

9.  What  is  ^  of  $144  ? 

1O.  How  many  twelfths  in  the  whole  of  any  sum  of 
money  ? 

DEFINITIONS. 

1.  A  Fraction  denotes  one  or  more  of  the  equal  parts 
of  a  unit. 

2.  A  fraction  is,  moreover,  the  expression  of  a  division 
that  has  not  been,  or  cannot  be,  performed. 

3.  The  dividend  is  called  the  Numerator,  and  is  always 
placed  above  the  line;  the  divisor  is  called  the  Denom- 
inator, and  is  always  placed  below  the  line. 

4.  The  numerator  and  denominator  are  called  the  terms 
of  the  fraction. 

5.  The  denominator  shows  into  how  many  equal  parts 
the  unit  or  single  thing  has  been  divided. 

6.  The  numerator  shows  how  many  equal  parts  form 
the  fraction. 

10 


146  ELEMENTARY  ARITHMETIC 

In  f ,  seven  is  the  denominator,  six  is  the  numerator,  and  6  and  7  are 
the  terms  of  the  fraction.      Can  the  division  be  performed  ? 

7.  In  a  Proper  Fraction  the  numerator  is  less  than  the 
denominator. 

f,  |,  f  £,  are  proper  fractions. 

The  value  of  a  proper  fraction  is  less  than  1.    Why? 

8.  In    an    Improper    Fraction   the    numerator   either 
equals  or  exceeds  the  denominator. 

7>  Tff>  if  >  are  improper  fractions. 

The  value  of  an  improper  fraction  equals  1  or  is  greater  than  1.    Why  ? 

9.  A   Mixed   Number    consists   of  an    integer   arid   a 
fraction. 

6f ,  20|,  31|i,  are  mixed  numbers. 

ORAL.   EXERCISES. 
1.  Analyze  the  fraction  -J^-. 

Explanation. 

1.  The  denominator  shows  that  the  unit  has  been  divided  into  12  equal 
parts. 

2.  The  numerator  shows  that  11  of  those  equal  parts  form  the  fraction. 

3.  The  fraction  is  read  "  eleven-twelfths." 

4.  11  and  12  are  the  terms  of  the  fraction. 

5.  As  dividend  and  divisor  they  denote  -^  of  11. 

2.  Analyze  in  like  manner  the  following : 

1-  *,  A.  it.  W.  *.  A.  A.  T*T,  A>  Ub  If.  tt,  A.  If • 

2-  A,  A.  A.  £ .  A.  tt>  A.  I.  A.  M.  4*.  W,  if  >  M- 

*•   T3 >  T9'  T3>  T2 '  "2T>  "3 ?  TJT>  T2 >  2~3 >  ¥2"'  4U>  "3~2 >  3T>  FOTT- 

3.  Express  by  Arabic  numerals  : 

1.  Sixteen  twenty-sevenths. 

2.  Thirteen  twenty-ninths. 

3.  Fifteen  thirty-seconds. 


FRACTIONS  147 

4.  Twelve  twenty-thirds. 

5.  Eighteen  twenty-fifths. 

6.  Nineteen  forty-seconds. 

7.  Eighty-one  ninetieths. 

8.  One  and  12  twentieths. 

9.  Two  and  three-ninths. 

10.  Nineteen  and  three  fortieths. 

11.  Six  sevenths.  26.  Eight  twenty-firsts. 

12.  Five  ninths.  27.  Eleven  twentieths. 

13.  Ten  fifteenths.  28.  Five  twenty-seconds. 

14.  Eight  twenty-thirds.  29.  Eight  thirty- sixths. 

15.  Three  eighteenths.  30.  Seven  twenty-fourths. 

16.  Five  twenty-fourths.  31.  Three  tenths. 

17.  Fifteen  thirtieths.  32.  Seven  fourteenths. 

18.  Eight  twenty-sixths.  33.  Six  twentieths. 

19.  Seven  forty-seconds.  34.  Eight  thirty-ninths. 

20.  Five  fourths.  35.  Eighteen  fortieths. 

21.  Seven  elevenths.  36.  Sixteen  seventeenths. 

22.  Five  eighths.  37.  Seven  twenty-firsts. 

23.  Seven  ninths.  38.  212  tenths. 

24.  Eight  twentieths.  39.  101  hundredths. 

25.  Five  thirteenths.  40.  Nine  and  five-sixths. 

4.  What  kind  of  numbers  and  fractions  are  : 

1.  f.        5.  tf,        9-  5f      13-  ¥&•      17-  rtW- 

2.  f.         6.  f.         10.  TV       14.  W-         18-  'rfr- 

3.  f         7.  6|.       11.  H-       15.  ff  19.  6f 

4.  y.       8.  f.         12.  f         16.  Tfo.         20.  m- 

5.  Change  the  following  Roman  numerals  to  Arabic, 
and  read  the  fractions  : 

V          IX          L        XL        III         VII         XIX 
X'       VIII'       C'       LX*       IV'       VIII'       XXI' 


148  ELEMENTARY  ARITHMETIC 

REDUCTION   OF  FRACTIONS. 

INDUCTIVE   STEPS. 

1.  Express  as  a  fraction  one  fourth  of  a  dollar. 

2.  Express  as  a  fraction  two  fourths  of  a  dollar. 

3.  Two  fourths  equal  how  many  half  dollars  ? 

4.  Write  the  equation,  one  half  a  dollar  equal  to  two 
fourths. 

5.  Since  ^  =  J,  how  can  the  terms  of  the  fraction  ^ 
be  changed  to  the  terms  of  the  fraction  |-  ? 

6.  How  can  the  terms  of  \  be  changed  to  the  terms 
off? 

7.  Multiplying  in  like  manner  by  2,  what  does  \  be- 
come? 

8.  How  can  -|  be  changed  back  to  \  ? 

9.  How  can  \  become  f  ?     How  can  f  become  \  ? 
1O.  Change: 

1.  \  to  8ths.  10.  ^  to  halves. 

2.  \  to  9ths.  11.  T3^-  to  fifths. 

3.  \  to  12ths.  12.  -£%  to  sixths. 

4.  |  to  halves.  13.  f  to  9ths. 

5.  f  to  thirds.  14.  f  to  8ths. 

6.  ^  to  thirds.  15.  f  to  16ths. 

7.  i  to  16ths.  16.  |  to  lOths. 

8.  1  to  15ths.  17.  ^|  to  halves. 

9.  |  to  18th.  18.  2^r  to  fourths. 

DEFINITIONS. 

1.  Reduction  changes  the  form  of  fractions  without 
changing  their  value. 


REDUCTION  OF  FRACTIONS  149 

2.  A  Common  Divisor  of  two  or  more  numbers  exactly 
divides  each  of  them. 

|  becomes  £  by  dividing  both  terms  by  their  common  divisor  3. 

3.  The    Greatest   Common   Divisor  of  two   or  more 
numbers  is  the  greatest  number  that  exactly  divides  each 
of  them. 

T\  becomes  £  by  dividing  both  terms  by  their  greatest  common  divisor,  6. 

4.  When  terms  have  no  common  divisor,  the  fraction 
is  said  to  be  in  its  Lowest  Terms. 

i.j  |?  i-£?  are  fractions  in  their  lowest  terms. 


PRINCIPLE. 

Multiplying  or  dividing  both  terms  of  a  fraction 
by  the  same  number  does  not  change  the  value  of 
the  fraction. 


Reduction  to  Lowest  Terms. 

EXERCISES. 

1.  Reduce  f$  to  its  lowest  terms. 
Process.  Explanation. 

6)_36.  — .    6    .  1.  According  to  the  principle,  we  must  divide  both 

2)  6    —  3  terms. 

2.  Dividing  by  6,  we  obtain  T%  ;  dividing  by  2,  we 
obtain  f . 

3.  The  terms  3  and  5,  having  no  common  divisor,  are  the  lowest  terms 
of  the  fraction  ££. 

4.  Hence,  ft  =  f . 

How  can  f  be  obtained  from  ||-  by  one  division  ? 
What,  then,  is  the  Greatest  Common  Divisor  of  36  and 
60? 


150  ELEMENTARY  ARITHMETIC 

RULE. 

1.  Divide  both  terms  of  the  given  fraction,  and  also  re- 
sulting terms,  by  any  common  divisor. 

2.  Continue  thus  to  divide  resulting  terms  until  terms 
are  found  that  have  no  common  divisor;  or, 

3.  Make  a  single  division  by  using  the  Greatest  Com- 
mon Divisor  of  the  given  terms. 

2.  Reduce  to  their  lowest  terms  : 

1.  |f.  11.  ||.  21.  Hi  31.  fff.  41.  ^. 

2.  if.  12.  ff.  22.  iff.  32.  ||f.  42.  IHf 

3.  it.  13.  28..  23.  iff  33.  ffi  43. 

4.  ££.  14.  ff.  24.  }ff  34.  fjf  44. 

5.  if.  15.  |f.  25.  Iff.  35.  |ff.  45.  fff 

6.  |f.  16.  ff.  26.  iff.  36.  f£f-  46.  .flft. 

7.  ff.  17.  ff.  27.  |ff.  37.  -Iff.  47. 

8.  Jf  18.  ||.  28.  fff.  38.  Tftfo.  48. 

9-  H-      19-  M-      29-  iff-      39-  To¥ir-     49-  T^ftV- 
10.  |f.      20.  ff.      30.  fff.      40.  ^     50.  ^Vj. 


Reduction  to  Higher  Terms. 

EXERCISES. 
1.  Reduce  f  to  56ths. 

Process.  Explanation. 

_i_  7  =  8.  !•  According  to  the  principle,  we  must  multiply 


5_  x  8  __  4_0.  o      terms. 

2.  56  H-  7  =  8,  the  multiplier  required. 

3.  Multiplying  both  terms  of  f  by  8,  we  obtain  |§,  a  fraction  in  higher 
terms.     Hence,  ^  =  |$. 


REDUCTION  OF  FRACTIONS  151 

2.  Reduce  to  higher  terms  : 

1.  £to!2ths.      15.  T56-to80ths.  29.  ^to88ths. 

2.  f  to  24ths.      16.  -^  to  102ds.  30.  f  to  88ths. 

3.  fto24ths.      17.  |ltol44ths.  31.  -^  to  88ths. 

4.  -&  to  50ths.    18.  ^  to  143ds.  32.  ^  to  576ths. 

5.  |lto48ths.    19.  -^tolSOths.  33.  |fto576ths. 

6.  ^  to  65ths.    20.  |f  to  216ths.  34.  i|  to  576ths. 

7.  2T  to  lOOths.  21.  ^5  to  300ths.  35.  ffi  to  576ths. 

8.  -^to42ds.     22.  ^  to  480ths.  36.  ^-to252ds. 

9.  ^-to45ths.    23.  ^_to341sts.  37.  f  to  252ds. 

10.  JL.  to  lOOths.  24.  -^-to399ths.  38.  -i-to252ds. 

11.  f  to  8ths.       25.  |  to  64ths.  39.  -^  to  68ds- 

12.  |f  to  121sts.  26.  H  to  169ths.  40.  ||  to  126ths. 

13.  |  to  125ths.   27.  f  to  200ths.  41.  f  to  lOOOths. 

14.  £  to  20ths.     28.  g  to  72ds.  42.  fgf  to  68ths. 


Reduction  to  Common  Denominator. 

1.  The  results  obtained  in  examples  21,  22,  and  23, 
having  the  same  denominator,  88,  are  called  Like  Frac- 
tions. 

2.  The   results   obtained   in   11,    12,   and   13,   having 
different  denominators,  are  called  Unlike  Fractions. 

3.  Hence,  Like  Fractions  have  the  same  denominator; 
Unlike  Fractions  do  not  have  the  same  denominator. 

4.  When  fractions  are  reduced  to  the  same  denomi- 
nator they  are  said  to  have  a  Common  Denominator. 

5.  When  the  common  denominator  is  the  least  one  ob- 
tainable, the  fractions  are  said  to  have  their  Least  Com- 
mon Denominator. 


152 


ELEMENTARY  ARITHMETIC 


1  .  Reduce  f  ,  f  , 
mon  denominator. 


EXERCISES. 
to  like  fractions  ;  that  is,  to  a  com- 


Process. 
3.  x  3  __  9 
4  x  4  __  16 

JL    4  =  :  JL 
TT  -  Tf' 


Explanation. 

1.  "We  cannot  but  see  that  by  reducing  f  and  f  to- 
12ths,  all  the  fractions  will  have  a  common  denomi- 


2.  Both  terms  of  f  multiplied  by  3  gives 


3.  Both  terms  of  f  multiplied  by  4  gives  £f  . 

4.  Hence,  the  results  required  are  T9j,  ^f  ,  ^. 

2.  Reduce  f  ,  -J,  -^  to  their  least  common  denominator. 


Process. 

L.  C.  D.  =  40 
40  _._     A  _  if) 

40  -r-  10  =     4. 


*  } 

X 


Explanation. 

1-  The  least  common  denominator  of  the 
given  fractions  must  be  the  least  number  that  4, 
8,  and  10  will  exactly  divide. 

2.  The  least  number  that  4,  8,  and  10  will 
exactly  divide  must  contain  the  least  number  of 
factors  that  will  produce  4,  8,  and  10. 


7.  X      5   __   35 
S 


_ 


3.  The  L.  C.  D.  =  2  X  2  X  2  X  5  =  40. 


Prime  Factors. 

10  =  2  x  5 
8=2x2x2 
4  =  2x2. 


4.  40  -4-  4,  8,  10  =  10,  5,  4,  the  multi- 
pliers required  to  reduce  the  given  fractions 

t°40ths- 

5.  Hence,  f  =  ft,  J  =  |$,  ^  =  |f . 

In  the  accompanying  table  of  factors  there 
are  but  two  different  factors, — 2  and  5.     In 
finding  the  L.  C.  D.  of  the  given  fractions, 
2  is  taken  three  times  and  5  is  taken  one»time,  which  is  the  greatest 
number  of  times  each  occurs. 

Hence,  the  L.  C.  D.  =  the  product  of  the  different  factors  of  the  given 
denominators  taken  the  greatest  number  of  times  that  each  occurs. 


REDUCTION  OF  FRACTIONS  153 

RULE. 

1.  Find  the  L.  C.  D.  of  the  given  fractions. 

2.  Reduce  the  fractions  to  the  L.  O.  D.  by  the  process 
for  obtaining  higher  terms. 

3.  Reduce  to  a  common  denominator : 

1.  f,  1,  H-  17-  A,  i,  A,  f 

2.  f,f,£  is-  f,«,A.A- 
3- 1,  A.  if-                19- 1.  f,  i.  f 

4.  |,f,f.  20.  l,|,M,f 

5-  I,  A,  A-  21.  i,  I,  A,  J. 

e.  I,  A.  A-  22.  A,  A,  I.  A- 

7.  f ,  I,  A-  23.  H,  A,  A,  A- 

8-  f  >  f ,  A-  24.  A,  A,  A.  H- 

9-  i,f,A-  25.  H.W.&.A- 

10.  A>  !>  A-  26.  «,  if,  A.  H- 

11.  I,  f,  f,  f  27.  A,  f,  if,  A- 

12.  |,  iV3^  if >  A-  ^S.  ^g,  -^-,  -jfy,  ^. 

13-  f,  i,  tf.  A-  29.  A,  A,  «,  A- 

14-  I,  «,  i,  1-  30.  i,  f ,  A,  A- 

is- 1,  A,  M,  i  si.  I,  A,  f,  f 

16.  f,  f,  f,  H-  32.  A,  I,  f,  i. 

Reduction  to  Mixed  Numbers. 

EXERCISES. 

1.  Reduce  ^f-  to  a  whole  or  a  mixed  number. 
Process.  Explanation. 

27  )  293  (  10|4  Division  is  indicated.     We  therefore  divide  293 

27  by  27,  and  obtain  lOff . 

Or,  we  may  say:   "1  =  ||;  therefore,  in  293 
twenty-sevenths  there  are  as  many  1's  as  27  is  con- 
00  tained  times  in  293,  or  lOff " 

23 


154  ELEMENTARY  ARITHMETIC 

RULE. 

Perform  the  division  indicated,  and  make  the  quotient 
exact. 

2.  Keduce  to  whole  numbers  : 

1.  f.         4.  *£..         7.  J»£.         10.  4A.         13.  2£. 

2.  y.       5.  J£.         8.  V-         11-  ¥•         14.  ff 

3.  if..       6.  A£.         9.  }.  12.  4£.         15.  -2/. 

3.  Reduce  to  mixed  numbers  : 

1.  £.       5.  -1/.       9.  Jgt.     13.  Y--     17-  ¥•     21-  ¥• 

2.  Y-     6-  V-     10-  ¥-•     14-  ¥•     18-  ¥•     22-  ¥•• 

3.  J£.     7.  Jj5-.     11.  -%6.     15.  ^L.     19.  ^.     23.  ff. 

4.  J^.     8.  %     12.  ZJL.     16.  J^.     20.  ^.     24.  45-. 

4.  Reduce  to  whole  or  mixed  numbers  : 

1.  iff.  5.  ifii.  9.  Hft.  13. 

2.  Jfli.  6.  if|i.  10.  i|p.  14. 
3-  -Vi1-  7.  ftf  11.  WJL.  15. 
4.  i.  8. 


Reduction  of  Mixed  Numbers. 

EXERCISES. 
1.  Reduce  57^  to  an  improper  fraction. 

Process.  Explanation. 

57  =  39.5.^  Since  1  =  |,  57  =  *94.     -3f-^  +  f  =  *%*. 

_39£     I     5  __  4  Q4  Or,  regarding  the  mixed  number  as  the  result 

of  a  division,  we  may  say:  "57,  the  quotient, 

multiplied  by  7,  the  divisor  =  399  ;  399  -f  5,  the  remainder  =  404,  the 
dividend.     Hence,  57f  —  ±$A." 


KEDUCTION  OF  FRACTIONS  155 

RULE. 

1.  Multiply  the  integer  by  the  denominator. 

2.  To  the  product  add  the  numerator. 

3.  Place  the  sum  over  the  denominator. 

2.  Reduce  to  improper  fractions  : 

1.  17f.          5.  86^.  9.  33&.  13.  56Qif 

2.  402H-   6.  17^V   10.  188&.  14.  256fjf 

3.  49£f    7.  29^.   11.  509^-.  15.  1307TV 

4.  62|f.    8.  93i}£.   12.  312^.  16. 


REVIEW. 

1.  Define  the  following  terms  : 

1.  Fraction. 

2.  Numerator. 

3.  Denominator. 

4.  Proper  Fraction. 

5.  Improper  Fraction. 

6.  Mixed  Number. 

7.  Reduction. 

8.  Lowest  Terms. 

9.  Common  Divisor. 

10.  Greatest  Common  Divisor. 

11.  Like  Fractions. 

12.  Unlike  Fractions. 

13.  Common  Denominator. 

14.  Least  Common  Denominator. 

2.  What  is  the  principle  of  reduction  to  higher  and 
lower  terms  ? 

3.  What  is  the  rule  of  reduction  to  lowest  terms  ? 

4.  What  is  the  L.  C.  D.  the  product  of? 

5.  What  is  the  rule  for  reduction  to  the  L.  C.  D.  ? 


156  ELEMENTARY  ARITHMETIC 

ADDITION    OF    FRACTIONS. 

INDUCTIVE    STEPS. 
1-  $i  +  $}  =  how  many  fourths  of  a  dollar  ? 

2.  £  of  a  day  and  ^  of  a  day  =  how  many  fifths  of  a 
day? 

3.  If  you  spend  -f$  of  a  dollar  for  pens  and  T77  of  a 
dollar  for  paper  and  ink,  how  many  tenths  and  how  many 
dollars  do  you  spend  ? 

4.  If  a  girl  reads  J  of  a  book  one  week,  J  of  the  book 
the  next  week,  what  fractional  part  of  the  book  has  she 
read? 

5.  Why  cannot  J  and  J  be  added  directly? 

6.  When  fractions  are  unlike,  what  change  must  be 
made  upon  them  before  they  can  be  added  ? 


PRINCIPLE. 

If  the   fractions  to  be   added  are  unlike,   they 
must  be  reduced  to  like  fractions. 


ORAL   EXERCISES. 

1.  Allen  gave  ^  of  his  money  for  oranges  and  f  for 
apples.     What  part  of  his  money  did  he  spend  ? 

2.  If  a  book  cost  f  of  a  dollar  and  some  paper  £  of  a 
dollar,  how  much  did  both  cost  ? 

3.  Mary,  having  %  of  a  dollar,  earned  ^  of  a  dollar. 
Had  she  then  more  or  less  than  one  dollar  ? 

4.  A  boy  spent  f  of  his  money  for  a  bat  and  £  of  it 
for  a  ball.     What  part  did  he  spend  for  both  ? 

5.  Three  ducks  cost  f  of  a  dollar  and  two  geese  y  of 
a  dollar.     What  was  the  entire  cost  ? 


ADDITION  OF  FRACTIONS  157 

6.  7^.  of  a  dollar  equals  how  many  dollars  ? 

Find  the  exact  quotient. 

7.  •§-£  of  a  dollar  equals  how  many  fifteenths  ? 

8.  I  sold  f  of  an  acre  of  land  to  one  man,  -fa  of  an 
acre  to  another,  and  ^  to  another.     How  many  acres 
did  I  sell? 

9.  A  man  spent  9J  dollars  for  a  coat  and  2f  dollars 
for  a  vest.     How  much  money  did  he  spend  for  both  ? 

Suggestion  :  9  +  2  and  £•  -f  £  . 

10.  9-J  dollars  equal  how  many  fifths  of  a  dollar  ? 

Analysis:  1  =  f  .     9  =  ^.     4A  +  f  =  V- 

11.  $2f  =  how  many  eighths?     Analyze. 

12.  Reduce  ^  to  a  mixed  number. 

13.  Eeduce  ^-  to  a  mixed  number. 

14.  Find  the  sum  of  9|  and  4J. 


"WRITTEN  EXERCISES. 
1.  What  is  the  sum  of  £,  f,  and  -J-? 

Process. 

1.  L.  C.  D.  =  2x2x2X3x5  =  120. 

2.  120-4-5,  6,8,  =  24,  20,  15. 

3.  t  +1+1 


Explanation. 

1.  The  fractions  are  unlike. 

2.  They  must  be  reduced  to  like  fractions. 

3.  L.  C.  D.  =2x2x2x3x5  =  120. 

4.  120  divided  by  5,  6,  and  8  give  24,  20,  and  15  as  multipliers  of  both 
terms.  96 

Hence,  $  =  jfr  ;  |  =  ^  J  1  =  if*-  100 

Adding,  we  have  f  {ft  =  2T\V  105 

301 


158  ELEMENTARY  ARITHMETIC 

2.  Find  the  sura  of: 

1-  I,  f  f          8-  4,  f,  1-  15.  f,  ft,  if 

2-  f  f ,  f          9-  f  A,  A-  16.  If,  2f ,  4A,  6|. 

3-  f  A.  H-    10-  f  f  if  I7-  1A.  2i  3i>  M- 

4.  f  A,  f      11.  A.  A,  A-  18.  2^,  5f,  i,  H. 

5.  i,  |,  M-      12.  f,  |,  A-  19.  5|,  -ft,  A,  12J. 

6-  i,  A.  A-    13-  *t.  5i'  i-  20- 

7.  |,  I,  A-      14-  3*.  6|,  3f.  21.  28A, 

3.  Find  the  sum  of  &,  ff, 


Process. 

2.  2  +  3  =  5. 
=  21,         3.  J  +  J  +  i=4-±l±-3=i|  = 

A  =  i-  4.  5  +  1A  =  6A- 

Explanation. 

1.  No  one  of  the  fractions  is  in  its  lowest  terms,  and  T|  is  an  improper 
fraction.     Reducing,  we  have  T6g-  =  ^,  f  f  —  2|,  -fa  =  £. 

2.  24-3  =  5,  the  sum  of  the  integers. 

3.  %  4-  \  4"  i>  reduced  to  a  common  denominator,  equals  1^. 

4.  5,  the  sum  of  the  whole  numbers,  plus  Ij1^,  the  sum  of  the  fractions, 
equals  6^. 

RULE. 

1.  Reduce  improper  fractions  to  whole  or  mixed  num- 
bers. 

2.  Reduce  all  fractions  in  higher  terms  to  then-  lowest 
terms. 

3.  Reduce  the  fractions  to  be  added  to  a  common  de- 
nominator. 

4.  Place  the  sum  of  the  numerators  over  the  common 
denominator. 

5.  Add   the   sum  of  the   integers   to   the   sum  of  the 
fractions. 


ADDITION  OF   FRACTIONS  159 

As  we  have  already  seen,  whole  numbers  occurring  should  be  added 
separately,  and  their  sum  finally  added  to  the  sum  of  the  fractions. 

4.  What  is  the  sum  of: 

1.  l  and  J  ?  8.  |  and  f  ?  15.  7^  and  8|  ? 

2.  |  and  J?  9.  4|  and  5f  ?  16.  £,  £,  and  J? 

3.  l  and  £  ?  10.  f  and  £  ?  17.  |,  J,  and  £  ? 

4.  l  and  l?  11.  |  and  l?  18.  fc  },  and  |? 

5.  land  i?  12.  f  and  f  ?  19.  f,  J,  andf? 

6.  |  and  1?  13.  6f  and  7f  ?  20.  If,  2|,  and  3J? 

7.  2|  and  3f  ?  14.  f  and  f  ?  21.  •£,  f,  and  f  ? 

5.  Find  the  sum  of: 

1-  M.  ft,  If.  i-  11.  2^,  8i,  27|,  9f 

2.  71  6&,  10f.  12.  f,  A,  A.  A»  «• 

3.  14f,  3A,  1*.  M-  13.  63A,  8f,  ¥>  ¥• 

4.  |,  1^,  lOf,  5.  14.  ITyAr,  AV.  «A. 

5.  2J,  |,  5f,  4f  16.  48|,  30,  63f,  8A, 

6.  2»A,  45,  16A-  16.  A,  A,  H,  T*T- 

7.  91i,  270J,  3A,  A-  17-  4F-.  ^  V- 

8.  23|,  6%  if1,  ¥•  18-  H-  7 

9-  ¥>  V,  ¥,  ¥>  I-       is-  H,  i 
10.  ¥,¥,«.*•  20.  ¥,  ¥,  H,  H,  *f 

"WRITTEN   PROBLEMS. 

1.  James    earned    $1-J,   John    $5^,    Asa    $11J,    An- 
drew $8J,  and  Thomas  $7^.     How  much  did  they  all 
earn  ? 

2.  A  man  walked   26J   miles  on  Monday,  13^  miles 
on  Tuesday,  21f  miles  on  Wednesday,  and  13^-  miles 
on    Thursday.      How    many    miles    did    he    travel    in 
all? 


160  ELEMENTARY  ARITHMETIC 

3.  A  man  bought  4  pieces  of  cloth.     The  first  con- 
tained 37f  yards;  the  second,  41f  yards;  the  third,  27f 
yards ;  the  fourth,  38^  yards.     How  many  yards  did  he 
buy? 

4.  If  a  pound  of  beef  costs  12f  cents,  a  pound  of  tea 
$1.21J,  a  pound  of  coffee  22|-  cents,  what  will  all  cost? 

5.  Two  sheep  cost  ty  of  a  dollar,  a  calf  f  of  a  dollar, 
and  a  lamb  f  of  a  dollar.     What  was  the  entire  cost  ? 

6.  A  man  who  had  spent  9|  dollars  for  a  coat  and  2} 
dollars  for  a  vest  had  6^  dollars  remaining.     How  much 
had  he  at  first  ? 

7.  Find  the  number  of  pounds  of  butter  in  4  tubs 
weighing  27J  pounds,  34f  pounds,  32£  pounds,  and  29f 
pounds. 

8.  I  gave  away  $38f  and  had  left  J5-J.     How  much 
money  had  I  at  first  ? 

9.  A  man  gave  his  three  boys  $5.00,  $7J,  and  $8J 
respectively,  and  had  $10^  left.     How  much  money  had 
he  at  first  ? 

10.  A  bicycler  rode  27|  miles  on  Monday,  33J  miles 
on  Tuesday,  37J  miles  on  Wednesday,  and  42£  miles  on 
Thursday.     How  far  did  he  ride  in  the  four  days  ? 

11.  A  three-sided  field  has  its  sides  31^,  46f, 
rods  long  respectively.     How  far  is  it  around  the  field? 

12.  Four  piles  of  wood  contain  37T%,  41-J,  29^, 
cords  respectively.      How  many  cords  are  in  the   four 
piles  ? 

13.  A  merchant  sold  46f  yards  of  cloth  for  $127^,  64| 
yards  for  $226f,  and  76^  yards  for  $312|.     How  many 
yards  did  he  sell,  and  how  much  money  did  he  receive  ? 

14.  Find  the  value  of  8J  +  }  +  H  + 


SUBTRACTION  OF  FRACTIONS  161 

SUBTRACTION  OF  FRACTIONS. 

INDUCTIVE  STEPS. 

1.  j|  —  $1  =  how  many  fourths  of  a  dollar  ? 

2.  £  of  a  day  —  f  of  a  day  =  how  many  fifths  of  a  day  ? 

3.  If  you  have  ^  of  a  dollar  and  spend  ^  of  a  dollar, 
how  many  tenths  of  a  dollar  have  you  left  ?     How  many 
fifths? 

4.  Some  boys  have  f  of  an  hour  for  play.     When  they 
have  played  J  of  an  hour,  what  part  of  an  hour  remains 
for  play  ? 

5.  Why  cannot  J  be  subtracted  from  £  directly  ? 

6.  When  fractions  are  unlike,  what  change  must  be 
made  upon  them  before  subtraction  ? 


PRINCIPLE. 

If  the  fractions  to  be  subtracted  are  unlike,  they 
must  be  reduced  to  like  fractions. 


ORAL  EXERCISES. 

1.  If  I  have  -f$  of  a  dollar,  and  spend  ^  of  a  dollar, 
how  much  will  I  have  left  ? 

2.  If  a  boy  has  f  of  a  dollar  and  gives  away  -|  of  a 
dollar,  how  much  money  has  he  remaining  ? 

3.  If  a  girl  who  has  f  of  a  dollar  gives  away  T3^  of  a 
dollar,  how  much  has  she  left  ? 

4.  Henry,  having  $-|,  gave  $^  for  a  slate  and  $  J  for  an 
inkstand.     How  much  money  had  he  left  ? 

5.  A  gentleman  owning  a  yacht   sold  -f^  of  it  to  a 
friend.     What  part  of  it  did  he  still  own  ? 

11 


162  ELEMENTARY  ARITHMETIC 

6.  A  boy  earned  $2£  and  immediately  spent  $£.    How 
much  had  he  then  ? 


Suggestion  :  f  2£  = 

7.  If  John  earns  J-J-  per  day  and  Henry  earns  $J-,  how 
much  more  does  John  earn  than  Henry  ? 

8.  If  a  man  earns  $3  per  day  and  his  wife  spends  $2J- 
thereof  per  day,  how  much  is  left  per  day  ? 

Suggestion  :  $3  =  $2f  . 

9.  I  spent  at  a  store  J-^  and  gave  in  payment  a  $2 
bill.     How  much  did  I  receive  in  change  ? 

10.  A  lady  having  bought  a  hat  for  f  5-J-  and  gloves  for 
,  received  how  much  in  change  out  of  a  $10  bill. 


WRITTEN   EXERCISES. 

1.  What  is  the  difference  between  f  and  ^-. 

Process.  Explanation. 

3    —  JL  .  1.  We    first    reduce    T\   to   its    lowest 

£  _  j^__£  _  2.  _  -3.  terms,  J  ;  and  then  write  f  —  £. 

2.  We  next  give  the  fractions  a  com- 
mon  denominator  by  reducing  \  to  |.     We  then  write  f  —  |  =  f  . 

2.  From  6f  take  2f       3.  Find  the  value  of  9f  —  3f 

Process.  Process. 

6|  =  m-  9*  =  9M  = 


2f  =  2if  3*  =  Stf  =  3}f. 


Explanation. 

1.  We  may  subtract  the  integers  and  fractions  separately. 

2.  Reducing  the  fractions  to  a  common  denominator  we  have  ^f  an 

3.  |§  cannot  be  subtracted  from  ^|. 

4.  1,  taken  from  9,  =  #  5  tt  +  tt  =  |f.  5.  8||  -  3ff  =  6ff  • 


SUBTKACTION  OF  FKACTIONS  163 

BULB. 

1.  Reduce  improper  fractions  to  whole  or  mixed  num- 
bers. 

2.  Reduce  all  fractions  to  their  lowest  terms. 

3.  Reduce  the  fractions  to  a  common  denominator. 

4.  Place  the  difference  of  the  numerators  over  the  com- 
mon denominator. 

5.  Add  the  difference  of  the  integers  to  the  difference  of 
the  fractions. 

4.  Find  the  value  of: 

1.  f-i-  22.  6f-2f 


3.  |  —  f  24.  18£  —  9f. 

4.  |  -  f  25.  13  -  6f 

5.  |  —  f  26.  22  —  8A- 

6.  A--f  27.  16f-10f. 

7.  6  -  4f  28.  |  +  i  -  i. 

8.  |-f  29.  |  +  f-i 

9.  A"f  30.  3f  +  i-2f. 
10.  A-i-  31.  i  +  f-f 

11.  ij-i  32.  f  +  ^-A. 

12.  A-A.  33.  t  +  i-A- 

13.  7|-3|.  34.  l-l-f^. 

14.  4-|.  35.  A—  i+> 

15.  7i~fr.  36.  4f  +2|-5i 

16-  A  -A-  37.  l  +  f  +  l-A- 

17.  A-A-  38.  6-2i  +  6i~A- 

18.  |f  —  A-  39.  5J-  +  3|  +  2^  --  10f 

19.  32f  ~  20f  40.  10|-  +  4Jf  —  6f  --  If. 

20.  44  —  16||.  41.  10  --  4|  +  12|  --  ^. 

21.  36|  —  18f  42.  8f  -f  4f  —  10J  —  A- 


164  ELEMENTARY  ARITHMETIC 

WRITTEN   PROBLEMS. 

NOTE. — Indicate  all  operations  before  performing  them. 

1.  A  farmer  having  4J  tons  of  hay  bought  8f  tons 
more.      He   then   sold   *l\  tons.      How  much    had  he 
left? 

2.  After  losing  $6^-  and  loaning  $3f ,  a  man  had  $18£. 
How  much  had  he  at  first? 

3.  A  horse  cost  $147f  and  a  carriage  $189^.     How 
much  more  did  the  carriage  cost  than  the  horse  ? 

4.  What  number   must  be  added  to  19^  to  make 
106f? 

5.  A  drover  sold  f  of  his  flock  of  sheep  to  one  man, 
and  \  of  it  to  another.     What  part  of  his  flock  did  he 
sell  ?     What  part  of  it  had  he  left  ? 

6.  A  man,  having  9|  dollars,  paid  3f  dollars  for  boots, 
and  4f  dollars  for  a  hat.     How  much  had  he  left? 

7.  The  sum  of  two  numbers  is  12^.     One  of  the 
numbers  is  7^.     What  is  the  other  number  ? 

8.  Add  together  f,  ^f-,  ^-,  and  from  their  sum  sub- 
tract ^. 

9.  -j%  of  a  pole  is  in  the  mud,  -£T  of  it  is  in  the  water, 
and  the  rest  of  it  is  in  the  air.     What  part  of  it  is  in 
the  air  ? 

10.  From  a  piece  of  silk  containing  30J  yards,  a  sales- 
man cut  off  3£  yards,  4f  yards,  and  12^  yards.     How 
many  yards  remained  ? 

11.  A  man  had  ^  of  his  sheep  in  one  pasture,  ^  in 
another,  and  ^  in  a  third,  and  the  remainder  in  a  fourth. 
What  part  is  in  a  fourth  ? 

12.  How  much  added  to  $10f  will  make  $15|. 


MULTIPLICATION  OF  FRACTIONS  165 

MULTIPLICATION  OF  FRACTIONS. 

INDUCTIVE    STEPS. 

1.  Two   times  f   equal   what?     Three   times  -|  equal 
what  ?     5  times  ^  equal  what  ?     •£$  =  what  in  lowest 
terms  ? 

2.  If  5  times   ^  =  •£,  how   may  the   ^  be   obtained 
directly  ? 

1.  Find  6  times  ^  in  lowest  terms  directly. 

2.  Find  8  times  -f±  in  lowest  terms  directly. 

3.  Find  10  times  -f$  in  lowest  terms  directly. 

3.  Is  it  multiplication  or  division  that  gives  lower  and 
lowest  terms  ? 

4.  There  are  two  ways  of  multiplying  a  fraction  by  an 
integer  : 

1.  Multiplying  the  numerator  multiplies  the  frac- 

tion. 

2.  Will  you  not  supply  the  second  way  ? 


PRINCIPLES. 

1.  Multiplying  the  numerator  multiplies  the  frac- 
tion. 

2.  Dividing  the  denominator  multiplies  the  frac- 
tion. 


To  Multiply  a  Fraction  by  an  Integer. 
ORAL   PROBLEMS. 

1.  If  one  book  costs  f  of  a  dollar,  what  will  6  books 
cost  ? 

2.  At  $|i-  per  yard,  what  will  9  yards  T)f  cloth  cost  ? 


166  ELEMENTARY  ARITHMETIC 

3.  How  much  will  1 6  books  cost  at  $£  each  ? 

4.  James  earns  $^  of  a  dollar  in  one  day.     How 
much  will  he  earn  in  12  days? 

5.  What  is  the  cost  of  15  bushels  of  oats  at  £  of  a  dol- 
lar a  bushel  ? 

6.  If  a  man  earns  $1|-  per  day,  how  much  does  he 
earn  in  6  days  ? 

7.  At  12^  cents  a  dozen,  what  will  6  dozen  oranges 
cost? 

8.  What  will   10  dozen  eggs  cost  at  15^  cents  per 
dozen  ? 

9.  What  is  the  cost  of  3  yards  of  cloth  at  $2f  a  yard  ? 
10.  If  6  times  6f  years  equal  twice  my  age,  how  old 

am  I? 

WRITTEN  EXERCISES. 

1.  Multiply  ^  by  9. 

Process.  Explanation. 

1    ^/  n  7        01  1-  Denominator  is  divisible  by  multiplier  9. 

}$  2  2.  Dividing  the  denominator  multiplies  the 

9 

fraction. 

3.  Cancelling  9  and  18,  we  have  |. 

4.  Reducing  |,  we  have  3^. 

2.  Multiply  12f  by  4. 

Process.  Explanation. 

123  =  ^  *  !•  Reducing  12|,  we  have  -6/. 

.6JL  \/  A. 252          <=in2  ^.  Denominator  5  is  not  exactly  di- 

3.  Multiplying  the  numerator  multiplies  the  fraction. 

4.  4  times  -«/  =  *$*  =  50|. 


MULTIPLICATION  OF  FRACTIONS  167 

RULE. 

Divide  the  denominator  or  multiply  the  numerator  of 
the  fraction  by  the  integer. 

NOTE.  —  The  integer  and  fraction  of  a  mixed  number  may  be  multiplied 
separately  and  the  results  added. 


3.  Multiply: 

1-  -j 

V  by  13. 

8.  $byll. 

15.  |f.  by  21. 

2.  , 

V  by  14- 

9.  44  by  25. 

lo       J 

16.  iffby  18. 

3.  { 

i  by  18. 

10.  |  by  400. 

17.  Mfby32. 

*'.! 

1  by  18. 

11.  2}  by  8. 

18.  |fi  by  60. 

5.  a 

^by7. 

12.  6|  by  5. 

19.  m  by  26. 

6.  ^  by  23.       13.  18f  by  21.       20.  tffa  by  144. 

7.  |  by  37.         14.  22|  by  108.     21.  ^^  by  273. 


"WRITTEN   PROBLEMS. 

1.  What  will  20  magazines  cost  at  $fa  each? 

2.  At  $-|  a  yard,  what  will  12  yards  of  cashmere  cost  ? 

3.  A  householder  bought  21  baskets  of  peaches,  each 
containing  f  bushels.     How  many  bushels  did  he  buy  ? 

4.  What  is  the  value  of  14  bushels  of  peaches  at  $-J 
per  bushel  ? 

5.  How  much  will  18  yards  of  silk  cost  at  $4f  per 
yard? 

6.  At  2|  cents  a  pound,  what  will  8  pounds  of  chalk 
cost? 

7.  If  a  family  consumes  5^  barrels  of  flour  in  1  year, 
how  much  would  they  consume  in  9  years  ? 

8.  When  flour  is  $8f  per  barrel,  how  much  must  be 
paid  for  16  barrels? 

9.  Find  the  weight  of  8  reams  of  paper  at  14^  pounds 
per  ream. 


168  ELEMENTARY  ARITHMETIC 

10.  Find  the  cost  of  81  acres  of  land  at  $28-£  per  acre. 

11.  Mr.  Brown  bought  21  tons  of  coal  at  $5J  a  ton. 
"What  was  the  cost  of  the  coal  ? 

12.  What  is  the  cost  of  27  cords  of  wood  at  $3£  per 
cord? 

13.  A  barrel  contains  31^-  gallons.     How  many  gallons 
in  13  barrels  ? 

14.  12  men  make  a  purchase  together,  each  paying 
$617f .     What  was  the  cost  of  the  purchase  ? 

15.  What  will  72  yards  of  cloth  cost,  at  $2.12£  per 
yard? 

16.  From  a  chest  of  tea  containing  45J  pounds,  14| 
pounds  are  sold.     How  many  pounds  remain  ?     What  is 
the  value  of  the  remainder  at  $1.00  per  pound  ? 

To  Multiply  an  Integer  or  a  Fraction  by  a  Fraction. 
INDUCTIVE   STEPS. 

1.  John  had  9  marbles  and   lost  ^  of  them.     How 
many  did  he  lose  ? 

9  X  ^  equal  how  many  ? 
9  X  J  =  how  many  ? 
j.  x  9  =  how  many  ? 

2.  Do  not  %  of  9,  £  X  9,  9  X  -J-,  9  times  J,  all  mean  the 
same  thing  ? 

3.  James  had  12  apples;  he  gave  away  J  of  them. 
How  many  did  he  give  away  ? 

How  many  ways  of  writing  the  operation  have  you  in 
mind  ? 

4.  What  does  the  word  "  of"  mean  in  such  cases? 

5.  What  sign  will  represent  "  of." 


MULTIPLICATION  OF  FRACTIONS  169 

6.  What  sign  stands  for  the  word  "  times." 

7.  f  of  15  means  what,  or  is  the  same  as  what? 

8.  Does  15  X  £  give  the  same  result  as  f  X  15  ? 

9.  In  f  X   15  what  numbers  may  be  cancelled  and 
what  is  the  result  ? 

1O.  Express  f  of  f  in  two  other  ways. 

Suggestion  :  Use  cancellation  and  state  your  result. 

WRITTEN  EXERCISES. 
1.  Multiply  7  by  £f 

Process.  Explanation. 

17  1.  7  X  H  means  H  of  7- 

17  x  7  _  W  _  17  _  4,  2.  A  of  7  =  A- 

9ft    ^    1  9«    ~  ~    A.    ~  ~       ** 

f  3.  ^  of  7  =  Y/. 

Q  •       4.  Cancelling  7  out  of  28  and  119 

we  have  -M-  =  4J. 
17        T        17 
—  X  T  =;:  -j-  =  4J.  In  practice  cancel  before  you  mul- 

4 


2.  Multiply  -f$  by  ^V 

Process. 

Explanation. 

1 

L 

2 

1 

x£_i 

K  2^  -  *• 

Cancelling  5  and 
the  product  is  £. 

9  we  have,  as  results,  above  the 
low  the  line,  2x4  =  8.    Hence, 

3.  Find  the  product  of  6J  and  4J. 

Process. 

6i  X  4J  =       X  J     ==      1  = 


Explanation. 

1.  Eeducing  the  mixed  number  to  improper  fractions  we  have  \3-  X  - 

2.  There  are  no  common  factors. 

3.  13  X  17  =  221  ;  2  X  4  =  8.  4.  The  product  is  *fi  or  27$  . 


170  ELEMENTAKY  ARITHMETIC 

GENERAL   RULE. 

1.  Express   integers   and   mixed  numbers   as   fractions 
and  indicate  the  multiplication. 

2.  Cancel  all  common  factors. 

3.  Form  products  of  the  resulting  numbers. 

What  is  the  principle  of  the  second  step  ? 

4.  Multiply : 

1.  14  by  f  4.  29  by  ^.  7.  24  by  10}. 

2.  24  by  |.  5.  106  by  £f .          8.  32  by  12J. 

3.  49  by  y.  6.  144  by  ff          9.  36  by  12f 

5.  Find  the  value  of: 

1.  7  x  A-  16-  I  of  55f 

2.  12  X  if  17«  T  of  3f 

3.  11  Xi-fr-  I8-  Tfof  H- 

4.  ££  X  11.  19.  ii  of  ff. 

5.  42|  X  9.  20.  fi  of  ||. 

6.  lOOf  X  12.  21.  2|  x  |  of  f 

7.  105  X  &.  22.  ty  x  1  X  A. 

8.  20f  x  48.  23.  ^  of  8J-  X  $. 

9.  5fJ  x  729.  24.  |  of  T\  of  ^. 

10.  81^  X  40.  25.  T5T  X  |  X  4f 

11.  |  of  «.  26.  if  X  5J  X  ^. 

12.  |  of  ^.  27.  ^  X  71  X  6f 

13.  H  of  ff.  28.  12J  X  A  X  16f 
14-  if  of  if.  29.  371X«XH- 
15.  |  of  65f  30.  8i  X  A  X  1TV 

6.  What  is  the  cost  of  : 

1.  |  of  a  ton  of  hay  at  $12|  per  ton  ? 

2.  32|  yards  of  broadcloth  at  $4|-  per  yard  ? 

3.  8  tons  of  coal  at  |5f  a  ton  ? 


MULTIPLICATION  OF  FRACTIONS  171 

4.  5J  cords  of  wood  at  $6f  a  cord  ? 

5.  f  of  an  acre  of  land  at  $100  per  acre  ? 

6.  18|  days'  work  at  $lf  per  day  ? 

7.  50^  pounds  of  sugar  at  4f  cents  per  pound  ? 

8.  lOf  yards  of  cloth  at  18f  cents  per  yard? 

9.  12  baseballs  at  $|  each  ? 

10.  87-|-  pounds  of  paper  at  3f  cents  a  pound  ? 

WRITTEN   PROBLEMS. 

1.  How  much  will  4^-  yards  of  cloth  cost  at  $!-£•  a 
yard? 

2.  At  $|-  per  bushel,  what  cost  21  baskets  of  peaches, 
each  containing  -§-  of  a  bushel  ? 

3.  24J  cubic  feet  equal  a  perch  of  stone.     How  many 
cubic  feet  in  5^-  perches  ? 

4.  A  man  having  387J-  acres  of  land  sold  £  of  it. 
How  many  acres  did  he  sell? 

5.  Reduce  to  simplest  form  f  of  -J  of  9^. 

6.  Mr.  A.  sold  |  of  150f  cords  of  wood  at  $10|  a 
cord.     What  did  he  receive  for  it  ? 

7.  Find  the  cost  of  28  hats  at  $2f  each. 

8.  Find   the   weight   of  8^   reams   of  paper  at   14f 
pounds  per  ream. 

9.  If  a  man  saw  3f  cords  of  wood  in  one  day,  how 
much  will  he  saw  in  -|  of  a  day  ? 

10.  A  horse  and  cow  were  bought  for  $180.     The  cow 
cost  \  as  much  as  the  horse.     Find  the  price  of  each  ? 

Suggestion  :  Let  |  =  the  price  of  the  horse. 

11.  5^-  yards  equal  1  rod.     How  many  yards  are  there 
in  fj-  of  8|  rods  ? 


172  ELEMENTARY   ARITHMETIC 

12.  A  grocer,  by  selling  4-|-  gallons  of  molasses  at  $-f  a 
gallon,  gains  $|.     What  did  the  4J  gallons  cost  him  ? 

13.  What  is  the  cost  of  15  boxes  of  starch,  each  con- 
taining 74-  pounds,  at  6J  cents  per  pound  ? 

DIVISION  OP  FRACTIONS. 

INDUCTIVE   STEPS. 

1.  $J  --  3  =  what?     |  of  a  ton  -f-  5  =  what?     ^  of 
a  mile  -j-  4  =  what  ? 

2.  In  dividing  by  3,  5,  and  4,  what  part  of  the  fractions 
did  you  divide  ? 

3.  What,  then,  is  the  effect  of  dividing  the  numerator  ? 

4.  One  half-dollar  equals  how  many  quarter- dollars? 
Then  $j- -s- 2  =  what?    $£ -*- 2  =  what?    $  of  a  ton -r- 

3  =  what  ?     ff-  of  a  mile  -*-  7  =  what  ? 

How  did  you  obtain  the  results  J,  -J-,  -^g-,  and  ^? 

5.  What,  then,  is  the  effect  of  multiplying  the  denomi- 
nator ? 


PRINCIPLES. 

1.  Dividing  the  numerator  divides  the  fraction. 

2.  Multiplying  the  denominator  divides  the  frac- 
tion. 


To  Divide  a  Fraction  by  an  Integer. 
ORAL   PROBLEMS. 

1.  If  5  slates  can  be  bought  for  $|f,  what  is  the  price 
of  each  slate  ? 

2.  If  5  pounds  of  sugar  cost  $^,  what  is  the  price 
per  pound? 


DIVISION  OF  FRACTIONS  173 

3.  A  farmer  received  $2f  for  8  bushels  of  oats.     How 
much  was  that  per  bushel  ? 

4.  I  paid  $67|-  for  coal.     If  I  bought  10  tons,  how 
much  was  that  per  ton  ? 

5.  How  many  barrels  of  apples  at  $2  per  barrel  can  be 
bought  for  $8}? 

6.  If  3  bushels  of  potatoes  are  sold  for  $f ,  what  is  the 
price  per  bushel  ? 

7.  A  laborer  earned  $15f  in  9  days.     How  much  was 
that  per  day  ? 

8.  If  4  pairs  of  shoes  cost  16£  dollars,  how  much  is 
that  per  pair  ? 

9.  I  paid  1^  dollars  for  5  pounds  of  butter.     How 
much  was  that  per  pound  ? 

10.  If  a  boy  can  walk  16f  miles  in  5  hours,  how  far 
can  he  walk  in  one  hour  ? 

"WRITTEN  EXERCISES. 

1.  Divide  £f  by  9. 

Process.  Explanation. 

1$  _._  «  2^          1.  The  numerator  18  is  divisible  by  9. 

31  31  2.  Dividing  the  numerator  divides  the  fraction. 

3.  Cancelling  9  and  18,  we  have  ^2T. 
Or,  we  may  say  :  "$f  -s-  9  =  £  of  £f ,  which  is  ^-." 

2.  Divide  ^  by  9. 

Process.  Explanation. 

§f-  -i-  9  =  |^-  =  1&.  1.  The  numerator  30  is  not  exactly  divis- 

ible by  9. 
2.  Multiplying  the  denominator  divides  the  fraction. 

3-  ¥  x  9  =  M  =  tf. 


174  ELEMENTARY  ARITHMETIC 

BULB. 

Divide  the  numerator,  or  multiply  the  denominator,  of 
the  fraction. 

3.  Divide: 

1.  |f  by  2.        11.  HI  by  5.  21.  25f  by  6. 

2.  if  by  6.        12.  ^ff  by  9.    .  22.  87|  by  12. 

3.  if  by  5.         13.  Iff  by  11.  23.  42f  by  18. 

4.  f-f-  by  7.        14.  iff  by  12.  24.  11  \  by  5. 

5.  |f  by  13.      15.  iff  by  13.  25.  33f  by  12. 

6.  if  by  15.      16.  f|f  by  18.  26.  174f  by  14. 

7.  ff  by  19.      17.  fff  by  16.  27.  878^-  by  15. 

8.  if  by  21.   18.  |ff  by  21.  28.  264|  by  7. 

9.  ||  by  20.   19.  fjf  by  341.'  29.  182f  by  9. 
10.  |f  by  25.   20.  ^  bJ  207-  30-  2697  bJ  144- 


•WRITTEN  PROBLEMS. 

1.  If  20  pencils  cost  $f  ,  what  part  of  a  dollar  is  the 
price  of  1  pencil  ? 

2.  If  7  dozen  eggs  cost  $f  ,  what  is  the  cost  per  dozen  ? 

3.  At  $5  per  yard,  how  much  silk  can  be  bought  for 
$18}  f 

4.  For  $221  how  many  yards  of  cloth  can  be  bought 
at  $3  per  yard  ? 

5.  A   father    divided    equally   among    his    5    children 
$478f  .     How  much  did  each  receive  ? 

6.  A.,  B.,  and  C.  gained  in  business  $734f.    Distribute 
the  gain  equally  among  them. 

7.  I  bought  25  pounds  of  butter  for  $5£.     How  much 
did  I  pay  per  pound  ? 

8.  If  54  horses  cost  $4622f  ,  what  is  the  cost  of  each  ? 


DIVISION  OF  FRACTIONS  175 

9.  A  man  paid  $99f£  for  4  cows.     How  much  was 
that  apiece  ? 

10.  If  9  men  consume  J  of  9f  pounds  of  meat  in  a  day, 
how  much  does  each  man  consume  ? 

11.  How  many  times  will  16f  gallons  of  cider  fill  a 
vessel  that  holds  3  gallons  ? 

12.  If    87   cows   cost  $3870^-,  what   is   their   average 
cost? 

13.  If  12  yards  of  silk  are  worth  f  16.46f ,  what  is  the 
value  per  yard  ? 

14.  John  can  walk  21  miles  in  |  of  a  day.     In  what 
part  of  a  day  can  he  walk  1  mile  ? 

15.  How  many  times  is  27  contained  in  |-  of  ^  of  42-J? 

16.  If  16  tons  of  hay  cost  $190f,  what  is  the  cost  of  1 
ton? 

17.  If  21  cords  of  wood  cost  $115J,  what  is  the  cost  of 
1  cord  ? 

18.  A  man  paid  $63f  for  15  tons  of  coal.     Find  the 
cost  per  ton  ? 

To  Divide  an  Integer  or  a  Fraction  by  a  Fraction. 

1.  To  reciprocate  is  to  give  in  return.     3  =  f .     The  re- 
ciprocal of  3  or  f  =  J,  so  called  because  the  space  below 
the  line  reciprocates  or  gives  in  return  3  for  3. 

2.  Since  f  X  %  =  1,  and  f  X  f  =  1,  and  so  on,  any 
two  quantities  whose  product  is  1   are  called  reciprocal 
quantities. 

3.  As  we  have  just  seen,  the  reciprocal  of  3  is  ^,  and 
the  reciprocal  of  f  is  f ,  or  •§•  inverted. 

4.  What  is  the  reciprocal  of  4,  5,  6,  8,  10,  12,  50,  144? 

5.  What  is  the  reciprocal  of  },  f ,  f ,  |,  -&,  f 


176  ELEMENTARY  ARITHMETIC 

WRITTEN  EXERCISES. 

1.  Divide  -J^-  by  f. 

Process.  First  Explanation. 

1 1    V  i  =  -|-5-  • — •  1-ft-I.  li  ig  tf16  dividend  ;  ^  the  divisor,  and 

the  quotient  is  to  be  found. 

If  1  or  ^  be  divided  by  ^  the  quotient  is  -£.  But  the  dividend  is  not  1, 
but  ^|  of  1,  and  therefore  the  quotient  required  is  ||  of  -£. 

j.£  Of  |  =   12X8  ==  ^  ~  ^^'  Q110^611^. 

Observe  that  in  j^-]  the  divisor  £  appears  in  inverted 
form  •£,  which  is  called  the  Reciprocal  of  ^. 
The  process  may  also  be  explained  as  follows : 

Second  Explanation. 

According  to  the  principle,  "  Multiplying  the  denominator  divides  the 
fraction,"  we  have  \\  -=-  5  =  121^  &.  Here  5,  the  numerator  of  the  frac- 
tion, is  used  as  a  divisor.  But  ^  means  5  divided  by  7.  Therefore,  the 
divisor,  5,  is  seven  times  as  large  as  it  should  be,  and  the  quotient  obtained 
by  dividing  by  5  is  only  one-seventh  as  large  as  the  true  quotient ;  hence, 
we  must  multiply  by  7.  12^  5  X  7  =  H^l  ==  ^  ==  1^'  (luotient- 

x 


Here,  again,  you  will  observe  that  in  \\  x  g,  the  divisor 
^  appears  in  inverted  form  \. 

Third  Explanation. 

Again,  both  dividend  and  divisor  may  be  reduced  to  a  common  denom- 
inator and  the  division  be  performed  thus  : 


From  the  foregoing  we  derive  the  following 

RULES. 

Invert  the  divisor  and  multiply.    Or, 

Reduce  dividend  and  divisor  to  common  denominator, 
and  then  divide  the  numerator  of  the  dividend  by  the  nu- 
merator of  the  divisor. 


DIVISION  OF  FRACTIONS 


177 


17-  6|  by  f. 

18.  THbyS 

19.  25  by  8 

20.  14ft  by 

21.  214|  by 

22.  ft  by  4 

23.  6|  by  8f 

24.  7|  by 


2.  Divide: 

1.  *  by  f          9.  A  ^  f 

2.  fbyf  10.  tfbyf 

3.  fby^.  11.  if  by  A- 

4.  f  by  f  12.  A  by  A- 

5.  A  by  f  .  13.  |f  by  f 

6.  A  by  ¥•     I4-  *i  by  A- 

7.  A  by  f-      15.  fj  by  A- 

8.  rf  by  f       16.  m  by  H- 

3.  Divide  f  of  £  of  f  by  f  of  {  of 

Process. 


Explanation. 

1.  Inverting  the  fractions  of  the  divisor, 

2.  Writing  X  throughout, 

3.  Cancelling  common  factors, 

4.  And  multiplying  together  remaining  factors, 

5.  We  have  -\%°  =  7£|. 

4.  Divide  •&  of  4  by  f  of  3J. 
Process. 


;0       1       5       13  ~~  325 
5 

Explanation. 

1.  4  and  3^  reduced  becomes  f  and  ^. 

2.  Inverting  divisor,  cancelling,  and  multiplying, 

3.  We  have  f  ff  == 


5.  Divide: 

1.  fof^byfofM-        2-  A  °f  H  by  $  of  if. 


12 


178  ELEMENTARY  ARITHMETIC 

8- t  of  f  of  if  by  f  of  A  of  H- 

4.  ^  of  H  of  If  by  ft  of  ft  of 

5-  HX&byjf  XTV 

6-  #  X  H  by  f$  -i-  if. 

7-  H  X  if*  ^  tf  X  #. 
8.  16|  by  18f. 

»•  f  X  f  X  f  by  f  X  i  X  6. 

10.  |  X  &  X  I  X  22  by  f  X  f  X  f  X  10. 

11-  |  X  f  X  f  X  3  by  |  X  |  X  H- 

12.  |  X  j-  X  £  X  9  by  |  X  f  X  f  of  2. 

13.  21  X  1ft  by  ^  X  A- 

14.  ^  of  3^  by  1ft  X  TV 

15.  |  of  A  by  ^  of  ^  of  4^. 

16.  |  of  |  of  A  by  |  of  i  of  f 
6.  Find  the  value  of: 

1.  f  of  3t  -f-  6J. 

2-  *  +  A  of  A- 

3-  (f-A)X-A. 
4.  (3f  -  3J)  X  }. 

6.  |  of  6J  -H  A  of  of 

6.  3J  of  8J  -s-  A  of  4|. 

7.  |  of  f  of  |f  H-  7. 

8.  f  of  4^  of  ^-  -4-  6^. 

9-  A  •«-  (¥  of  3i  of  2f). 

10.  A  -*-  (f  x  3^  x  2f). 

11.  (|--f)  x  I^T. 

12.  (f  of  |  of  5J)  -i-  (if  of  48). 

13.  (5J  +•  18|)  X  (11H  •+•  12A> 

14.  (f  of  2|V  2J)  X  |  H-  (i  of  |). 

15.  (|  of  |  of  2£)  •*-  (j.  of  3|). 

16.  (7|  -*-  4f)  H-  f  of  |  of  (5|  *.  8f). 


DIVISION  OF  FRACTIONS  179 


17.  (311  +  8)  X  (6f 

18.  21  X  f  X  (21  -^  |)  of  if 
.         19.  (31  +  41)  X  V  X  (I  -  3 

20.  (3|  of  1  A)  -*-  (I  of  6|). 


WRITTEN   PROBLEMS. 

1.  I  paid  $12   for  baseballs,  at  f  of  a  dollar   each. 
How  many  did  I  buy  ? 

2.  At  J  of  a  dollar  a  pound,  how  much  butter  can  be 
bought  for  ^|-  of  a  dollar  ? 

3.  If  a  man  pays  $1  J  per  day  for  his  board,  for  how 
long  a  time  will  $25  pay  for  his  board  ? 

4.  How  many  times  can  a  jar  holding  J  of  -J  of  a 
gallon  be  filled   from   another  jar   containing   f  of  3J 
gallon  ? 

5.  At  $3f  a  cord,  how  many  cords  of  wood  can  be 
bought  for  $40  ? 

6.  At  $-f  a  pound,  how  many  pounds  of  butter  will 
$110  buy? 

7.  At  $6^  a  bushel,  how  many  bushels  of  clover-seed 
can  be  bought  for  $40f  ? 

8.  How  many  books,  at  $3^-  per  volume,  can  be  pur- 
chased for  $31  J? 

9.  If  -ft  of  an  acre  of  land  is  worth  $23f,  what  is  one 
acre  worth  ? 

10.  How  many  sheep  must  I  sell  at  $3^  a  head  to  obtain 
$169? 

11.  If  a  yard  of  silk  costs  $2^,  how  many  yards-  can  be 
bought  for  $18^? 

12.  If  a  man  earns  $lf  a  day,  in  how  many  days  will 
he  earn  $12J? 


180  ELEMENTARY  ARITHMETIC 

13.  I  paid  $38J  for  g£  yar(js  of  cioth.     What  was  the 
price  per  yard? 

14.  How  many  pounds  of  butter,  at  32J  cents  a  pound, 
must  be  given  for  37}  pounds  of  sugar,  at  6  cents  a 
pound  ? 

15.  By  what  must  2f  -r-  3£  be  multiplied  to  give  the 
product  1  ? 

16.  If  a  man  travel  28^  miles  in  one  day,  how  many 
days  will  it  take  him  to  travel  177f  miles? 

17.  453J  -5-  5£  x  11  i  =  what? 


COMPLEX  FRACTIONS. 

1.  Reduce  |  to  a  simple  fraction. 

$ 

Process. 


Explanation. 

1.  |  is  dividend  and  |  divisor. 

2.  We  invert  the  divisor  and  multiply. 

3.  The  product  is  ff  ,  or  1^. 

2.  Reduce  -f  to  a  simple  fraction. 

Process. 
2-*-6    =     -5-Jy-  =  t  X 


Explanation. 

1.  2£  is  dividend,  6£  divisor. 

2.  Reducing  the  mixed  numbers,  inverting  the  divisor,  and  multiply- 
ing, we  have  T3/j- 


FRACTIONAL  RELATION  181 

3.  Find  the  value  of: 
if  7    HI 

'  T  T* 

T"  '"  *F 

0     54  Q    41 

I  5f 

ios 
10.  ±z. 


>.  1.  11.  U|-i 

j    a  12.  1^1.  18.   H-M. 

8f  |of4i  ttf-*-B 


FRACTIONAL    RELATION. 

INDUCTIVE    STEPS. 

1.  $5  is  J  of  how  many  dollars?     $10  is  J  of  how 
many  dollars  ?     $20  is  £  of  how  many  dollars  ?     $20  is 
£  of  how  many  dollars  ?     $20  is  £  of  how  many  dollars  ? 

2.  Of  what  number  is  20  four-fifths  ?     8  is  f  of  what 
number  ?     8  is  -|  of  what  number  ? 

3.  Since  8  is  f  of  12,  f  expresses  the  relation  of  8  to  12. 

4.  Since  8  is  J-  of  9,  f  expresses  the  relation  of  8  to  9. 

5.  8  is  f  of  what  number?     Since  8  is  f  of  12,  and 
since  |-  X  f  =  12,  how  may  the  required  number  be 
found  ? 


PRINCIPLE. 

The  given  number  is  the  dividend. 
The  fraction  of  relation  is  the  divisor. 
The  required  number  is  the  quotient. 


132  ELEMENTARY  ARITHMETIC 

EXERCISES. 

1.  84  is  f  of  what  number  ? 

Process.  Explanation. 

14  1.  84  is  the  dividend. 

84  _^_  6  —  $£  v  1  —  QQ  2.  f  is  the  divisor. 

i    *  7  —  i       i  — 

3.  The  quotient  is  required. 
4.   Y-  X  I  =  ¥-  X  7  =  98. 
Or,  we  may  say,  "  f  of  the  number  =  84,  \  =  14,  $  =  98." 

RULE. 
Multiply  by  the  fraction  of  relation  inverted. 

2.  To  apply  the  rule : 

1.  18  is  -J  of  what  number? 

o 

a.  The  fraction  of  relation  inverted  is  f . 

b.  ¥•  X  |  =  ^  =  27. 

2.  60  is  ^  of  what  number  ? 

3.  125  is  f  of  what  number? 

4.  216  is  ff  of  what  number? 

3.  To  apply  analysis : 

1.  47  is  |  of  what  number? 

|  of  number  =  44  ;  l  of  number  =  22 ;  f  of  number  =  66. 

2.  45  is  f  of  what  number  ? 

3.  |  of  |  is  %  of  what  number  ? 

4.  1-|  is  4  of  ^  of  what  number  ? 

o          o  o 

WRITTEN  PROBLEMS. 

1.  If  $64  is  f  of  my  money,  how  much  money  have  I? 

2.  Sold  a  watch  for  $43|,  which  was  -J  of  its  cost. 
What  did  it  cost? 

3.  What  is  the  price  of  land,  per  acre,  when  -^j-  of  an 
acre  costs  $44.25. 

4.  If  $425  is  ^1  of  my  salary,  what  is  my  salary  ? 


FRACTIONAL  RELATION  183 

5.  Sold  my  farm  for  $3360,  which  was  f  of  its  value. 
Find  its  value. 

6.  A  house  was  sold  for  £  of  its  cost.     If  the  selling 
price  was  $2100,  what  was  the  cost? 

7.  A  freight  train  ran  15  miles  per  hour,  which  was  f 
as  fast  as  an  express  train.     What  was  the  rate  of  the 
express  train  ? 

8.  If  $6000  is  f  of  the  value  of  my  farm,  what  is  f 
of  its  value  ? 

9.  If  $96  is  the  cost  of  f  of  an  acre,  what  will  one 
acre  cost? 

10.  If  f  of  a  box  of  oranges  cost  $5.50,  what  will  one 
box  cost  ?     8  boxes  ? 

11.  J   of  a   foundry   is   worth   $540}.      What   is   the 
foundry  worth  ? 

12.  If  |-  of  an  acre  of  land  cost  $80,  what  will  1  acre 
cost  ?     What  will  -^  cost  ? 

13.  f  of  a  barrel  of  flour  costs  $4.20.     Find  the  cost 
of  a  barrel. 

14.  -fT  of  5f  is  y^  of  what  number  ? 

15.  A  man  sold  |-  of  his  share  of  stock  for  $5120. 
What  was  his  share  worth?     If  he  owned  -fa  of  the 
stock,  what  was  the  stock  worth  ? 

16.  If  YV  of  a  task  can  be  done  in  |-  of  a  day,  in  what 
time  can  the  whole  task  be  performed  ? 

17.  If  -f-  of  a  box  of  pens  costs  25  cents,  what  do  18 
boxes  cost? 

18.  Mr.  Brown  sold  a  horse  for  f  of  its  cost,  and  re- 
ceived $75. .    What  was  the  cost  of  the  horse  ? 

19.  A  farmer  sold  ^  of  his  sheep,  keeping  20.     How 
many  sheep  had  he  at  first  ?     How  many  did  he  sell  ? 


184  ELEMENTARY  ARITHMETIC 

20.  A  man  bequeathed  to  his  wife  $36,000,  which  was 
-fr  of  his  estate.     The  remainder  was  divided  equally 
among  his  4  children.     What  did  each  child  receive  ? 

21.  If  f  of  the  gain  equals  -f%  of  the  cost,  what  part  of 
the  cost  does  the  whole  gain  equal  ? 


ORAL   REVIEW. 

1.  Find  the  value  of  -J-  +  f  -f  }. 

2.  Find  the  sum  of  £,  J,  £. 

3.  In  $6f  how  many  fourths  of  a  dollar  are  there? 

4.  Eeduce  to  improper  fractions  :  ?•§-,  6f  ,  5-f  ,  7|, 

5.  Reduce  to  integers  or  mixed  numbers  :  J^, 


6.  What  is  the  sum  of  6J,  4f,  7£,  4J-? 

7.  "What  is  the  value  of  $42£  —  $39£. 

8.  Find  the  difference  between  20  acres  and  6-J  acres. 

9.  If  a  man  spends  ^  of  his  money,  what  fraction  of 
it  has  he  left  ? 

10.  I  spent  $16J  and  had  $11|  left.    How  much  money 
had  I  at  first  ? 

11.  If  $36  is  £  of  what  an  article  cost,  what  did  the 
article  cost  ? 

12.  A  man  sold  a  watch  for  $70,  which  cost  him  only 
^  of  that  sum.     How  much  did  he  gain  by  the  sale  ? 

13.  What  will  3  pairs  of  shoes  cost  at  $3|  a  pair? 

14.  If  a  horseless  carriage  runs  60^  miles  in  6  hours, 
at  what  rate  per  hour  does  it  run  ? 

1  5.  A  farmer  bought  a  horse  and  a  cow.  The  cow  cost 
him  $30.  f  of  this  sum  is  \  of  what  he  paid  for  the 
horse.  What  did  he  pay  for  the  horse  ? 


FRACTIONAL  RELATION  185 

16.  If  f  expresses  the  relation  of  •£  to  some  other  num- 
ber, what  is  that  number  ? 

17.  66f  is  f  of  what  number? 

18.  |  of  a  number  -f-  ^  of  f  of  the   number  =  40. 
What  is  the  number  ? 


WRITTEN   REVIEW. 
1.  Find  the  sum  of  1-&,  7^,  f  . 


2.  A  man  having  $27f  received  $16T9Q-  for  work,  and 
paid  out  $18-|-.     How  much  had  he  then  ? 

3.  A  boy  has  $3^  to  buy  a  dog  worth  $5|-.     How 
much  more  money  must  he  get  ? 

4.  How  much  will  6  men  earn  in  6J  days  at  $3J  each 
per  day  ? 

5.  A  person  spending  ^,  J-  ,  and  ^  of  his  money  had 
$119  left.     How  much  had  he  at  first? 

6.  How  much  will  2f  dozen  eggs  cost  at  12J  cents  a 
dozen  ? 

7.  A  farmer  sold  22-J-  pounds  of  butter  at  -^  of  a  dol- 
lar a  pound,  and  took  in  exchange  cloth  at  f  of  a  dollar 
a  yard.     How  much  cloth  did  he  get  ? 

8.  How  many  books  at  66f  cents  each  can  be  bought 
for  $36. 

9.  What  is  the  value  of  54  loads  of  wheat,  each  con- 
taining 25  bushels,  at  $1  J  per  bushel  ? 

10.  After  spending  £  of  his  money,  a  man  had  $36  re- 
maining.    How  much  had  he  at  first  ? 

11.  A  farmer  buys  a  horse  for  $140,  and  sells  it  at  an 
advance  of  ^j-  of  the  cost.     What  is  the  selling  price  ? 

12.  Find  the  value  of  *  °^  %°L-' 

i  01  3| 


186  ELEMENTARY  ARITHMETIC 


13.  The  sum  of  two  numbers  is   12^.     One  of  the 
numbers  is  7-f-.     What  is  the  other  number  ? 

14.  A  grocer,  having  -J  of  a  barrel  of  sugar,  sold  £  of 
it  for  4|  dollars.     What  was  the  value  of  the  barrel  at 
the  same  rate  ? 

15.  A  man  owns  87T5g-  acres  of  land,  his  wife  owns  42f 
acres,  and  his  son  29f  acres.     How  many  acres  do  they 
own  together  ? 

16.  Of  a  certain  farm,  ^  is  pasture,  -J  is  under  cultiva- 
tion, and  the  remainder  is  woodland.     If  the  woodland 
is  50  acres,  how  many  acres  in  the  whole  farm  ? 

17.  A  wind  blowing  28-$-  miles  an  hour  blows  how  far 
in  10J  hours  ? 

18.  If  A.  can  do  a  piece  of  work  in  4  days,  how  much 
of  it  can  he  do  in  1  day?     If  B.  can  do  a  piece  of  work 
in  5  days,  how  much  can  he  do  in  1  day  ?     How  much 
can  A.  and  B.  do  together  in  1  day  ? 

19.  If  A.  and  B.  can  do  J  -f-  -J-  of  the  work  in  one  day, 
in  what  time  can  they  together  do  the  whole  work  ? 

20.  Two  men  together  earned  $870.     If  one  earned  f 
as  much  as  the  other,  how  much  did  each  earn  ? 

21.  What  number  divided  by  ^  equals  6^?     What 
principle  is  involved  ? 

22.  At  $2^-  per  barrel,  how  many  barrels  of  apples  can 
be  bought  for  $55  ? 

23.  A  dealer  in  farming  implements  paid  $149^  for  8 
ploughs.     For  how  much  apiece  must  he  sell  them  to 
gain  $8-|  on  each  plough  ? 

24.  A  man  bought  14  tons  of  hay  at  $12|-  a  ton,  and 
sold  it  at  $16f  a  ton.     How  many  dollars  did  he  gain? 
How  many  dollars  did  he  gain  per  ton  ? 


FRACTIONAL  RELATION  187 

25.  How  many  bushels  of  oats  at  J-  of  a  dollar  a  bushel 
will  pay  for  f  of  a  barrel  of  flour  at  $4f  a  barrel  ? 

26.  When  wheat  sells  at  $1-|-  per  bushel,  how  many 
bushels  can  be  bought  for  $198  ? 

27.  From  -I  of  I-  of  -J  of  34  subtract  4-  of  -§-,  and  reduce 

y  o  o  o  *  IF' 

to  lowest  terms. 

28.  If  J  of  a  pound  of  tea  costs  $-|-,  how  many  pounds 
can  be  bought  for  $7|  ? 

29.  Find  the  difference  between  3|  X  6|  and  7|  -=-  If. 

30.  A  man  sold  l  and  ^  of  his  farm,  and  had  26f  acres 
left.     How  many  acres  had  he  at  first  ? 

31.  What  will  75  men  earn  in  18f  days,  if  each  earns 
^A  dollars  each  day  ? 

32.  If  it  takes  1 1  men  45f  days  to  do  a  piece  of  work, 
how  many  days  will  it  take  1  man  to  do  the  same  work? 

33.  Reduce  to  its  lowest  terms  j?|||. 

34.  Reduce    *^  *  to  a  simple  fraction. 

35.  If  3  be  added  to  both  terms  of  the  fraction  4.  will 

o " 

the  value  be  increased  or  diminished  ? 

36.  |  of  72  is  |  of  what  number? 

37.  Find  the  sum  of  2^-,  4|,  3f 

38.  Find  the  value  of  728  —  f  —  |  —  f  —  £. 

39.  How  long  is  a  post  of  which  5  feet  is  above  water, 
|  is  in  the  water,  and  -J  in  the  mud  ? 

40.  Reduce  to  its  simplest  form  *  *     71  ,  ^l/". 

4£  —  1|  -)-  2^ 

41.  Reduce  4M4  to  its  lowest  terms. 

O   A  O  O 

42 .  Find  the  smallest  number  that  will  exactly  contain 
15,  18,  21,  24,  and  30. 

43.  Find  the  prime  factors  of  1226,  1938,  and  2346, 
and  also  the  G.  C.  D.  of  these  numbers. 


188  ELEMENTARY  ARITHMETIC 

44.  Find  the  cost  of  8  rolls  of  carpet,  42J  yards  in  a 
roll,  at  91f  cents  per  yard. 

45.  A  horse  and  cow  were  bought  for  $180;  the  cow 
cost  %  as  much  as  the  horse.     Find  the  price  of  each. 

46.  If  |  of  $  of  a  ship  cost  $70,000,  what  is  ^-  of  it 
worth  ? 

47.  Divide  ~  —  |  by  -£-. 

48.  A.  and  B.  can  do  a  piece  of  work  in  10  days.     A. 
can  do  ^  as  much  as  B.     In  what  time  can  each  do  the 
work? 

49.  From  f  of  ^  take  ^  of  If 

50.  Find  the  value  of  MDCCCXCIX  -f-  ^. 

51.  Find  the  value  of: 

1.  (16|  +  14^)  X  f 

2-  (I  of  T6T  +  15)  X  (15  -  f  of  f). 

3-  (If  X  «)  -*-  («  X  tf  ). 

4-  (t  +  A)  X  T3T  +  (A  +  *)  X  3. 

5.  (|  of  8J  -3  X  A)X(t-s-i+H)- 

6.  (A  -  A  -  TV  +  T4y)  +  281. 

4J  -  3f  +  12^)  X  70. 

2f 
10 


52.  What  is  the  principle  of  : 

1.  Addition  of  fractions?     "What  is  the  rule? 

2.  Subtraction  of  fractions  ?     What  is  the  rule  ? 

53.  What  are  the  principles  of  : 

1  .  Multiplication  of  fractions  ?    What  is  the  rule  ? 

2.  Division  of  fractions  ?     What  is  the  rule  ? 

3.  Fractional  relation  ?     What  is  the  ru4e  ? 


DECIMAL  FRACTIONS  189 

DECIMAL  FRACTIONS. 

INDUCTIVE   STEPS. 
1.   1  ten  =  10. 

1.  How  often  does  10  occur  as  a  factor  of  100  ?    Then 

100  =  10  X  10. 

2.  How  often  does  10  occur  as  a  factor  of  1000? 

Then  1000  =  what? 

3.  How  often  does  10  occur  as  a  factor  of  10,000? 

Then  10,000  =  what? 

2.  Decimal  [Latin,  decent,  ten]  means  consisting  of  tens. 

A>  T*TT»  TTOT>  TWGT>  etc->  are  called  Decimal  Fractions 
or  Decimals,  on  account  of  their  decimal  denominators. 

3.  Decimals  are  fractions  having  1  with  ciphers  an- 
nexed for  their  denominators. 


2-  A  of  TJhr  =  what? 
3.  TVof  1000  =  what? 

4.  What  law  of  increase  and  decrease  governs  both 
Decimals  and  Integers  ? 

5.  Since  10  of  any  order  of  decimals  equal  1  of  the 
next  higher  order,  the  denominator  of  a  decimal  may  be 
indicated  by  the  position  of  the  numerator. 

6.  The  numerator  is  always  preceded  by  a  mark  [  .  Jj 
called  the  Decimal  Point. 

1.  y1^  =  .1.     First  place. 

2.  y^j-  —  .01.     Second  place. 

3.  T^  =  .001.     Third  place. 

4.  T^Tnj-  =  .0001.     Fourth  place. 

5.  One  and  one-tenth  is  written  1.1. 

6.  One  and  one-hundredth  is  written  1.01. 


190 


ELEMENTARY  ARITHMETIC 


7.  The  decimal  point,  being  thus  used  to  separate  the 
integer  and  decimal  of  a  number,  is  called  the  Separatrix. 

1.  .5  =  y5^,  read  "  five  tenths." 

2.  .05  =--  yfo,  read  "five  hundredths." 

3.  .005  =  y^j-,  read  "  five  one-thousandths." 

4.  .0005  =  -nHhnj-,  read  "  five  ten-thousandths." 

5.  .00005  —  1 6  050  0  g-,  read  "  five  hundred-thousandths." 

6.  .000005  =  TTjHhnrr*  read  "  five  millionths." 

Fix   in    your    mind   that   6    decimal    places    express 
millionths. 

Numeration  Table. 


1 

03 

R 
S 

-thousandths. 

"I 

02 

1 

§ 

0 

£ 

"S 

1 

w 

S 

S 

.-§ 

*c 

r^ 

1 

d 

3 
W 

§ 

e 

i 

H3 

C 
W 

.2 
1 

3 

V-«_ 

4 

5   . 

•> 

6 

3 

2 

i 

7 

4 

INTEGERS. 

DECIMALS. 

The  orders  below  millionths  are :  Ten-millionths,  hun- 
dred-millionths,  billipnths,  ten-billionths,  hundred  bill- 
ionths,  etc. 

In  the  table,  what  place  is  held  by : 


1.  Tenths? 

2.  Millionths? 

3.  Hundredths? 

4.  Thousandths? 

5.  Ten-thousandths? 


6.  Millionths? 

7.  Hundred-thousandths  ? 

8.  Ten-thousandths? 

9.  Hundredths? 
10.  Tenths. 


DECIMAL  FRACTIONS  191 

In  the  table,  what  is  the  decimal  name  of 'the : 

1.  First  place?  7.  Sixth  place? 

2.  Sixth  place  ?  8.  Fifth  place  ? 

3.  Second  place  ?  9.  Fourth  place  ? 

4.  Fifth  place?  10.  Third  place? 

5.  Third  place?  11.  Second  place? 

6.  Fourth  place?  12.  First  place? 
5.6  is  read  "  Five  and  6-tenths." 

45.63  is  read  "Forty-five  and  63-hundredths." 

345.632  is  read  "  Three  hundred  forty-five  and  632- 
thousandths." 

The  decimal  point  is  read  "  and." 

In  reading  a  decimal,  only  the  decimal  name  of  the 
last  figure  is  given. 

EXERCISES. 
1.  Read  53.467. 

1.  The  integer  is  read  "  Fifty-three." 

2.  The  point  is  read  "  and." 

3.  467  is  read  "  Four  hundred  sixty-seven  thousandths." 

4.  53.467  is  read   "Fifty-three  and  four  hundred  sixty-seven 

thousandths." 

NOTE. — When  there  is  no  integer  the  point  is  not  read. 

RULE. 

1.  Read  the  decimal  as  you  read  an  integer. 

2.  Close  with  the  decimal  name  of  the  right-hand  figure, 

2.  Read: 

1.  .38.  6.  3.2.  11.  300.45. 

2.  .58.  7.  4.03.  12.  126.567. 

3.  .487.  8.  5.004.  13.  75.890. 

4.  .056.  9.  6.0005.  14.  87.0781. 

5.  .0579.  10.  7.00006.  15.  999.00089. 


192  ELEMENTAEY  ARITHMETIC 

16.  .5346.  21.  81.000007.  26.  1000.321467. 

17.  .7935.  22.  92.123456.  27.  5000.000078. 

18.  .80465.  23.  100.789012.  28.  6789.000005. 

19.  .915766.  24.  246.345678.  29.  1234.123456. 

20.  .0268778.  25.  757.009102.  30.  5678.901234. 

3.  Write  169  thousandths  as  a  decimal. 

1.  Writing  the  given  number  as  an  integer,  we  have  169. 

2.  Prefixing  the  decimal  point,  we  have  .169. 

4.  Write  as  a  decimal  36  ten-thousandths. 

1.  Writing  the  number  as  an  integer,  we  have  36. 

2.  Ten-thousandths  occupy  the  fourth  place. 

3.  Prefixing  two  ciphers,  we  have  0036. 

4.  Prefixing  the  decimal  point  to  that  result,  we  have  .0036. 

RULE. 

1.  Disregarding  the  decimal  name,  write  the  given  num- 
ber as  an  integer. 

2.  When  necessary,  prefix  ciphers  to  give  the  last  digit 
the  decimal  name  required. 

3.  To  the  result  prefix  the  decimal  point. 

5.  Write  as  decimals  the  following : 

1.  Eight  tenths.     Seven  tenths.    Six  tenths.    One 

tenth. 

2.  Twenty-five     hundredths.      Thirty-two     hun- 

dredths. 

3.  Twenty-seven    thousandths.     Three    hundred 

three  thousandths. 

4.  Eight  ten-thousandths.     Ninety-five  hundred- 

.    thousandths. 

5.  Eight  and  three  hundred  seventeen  millionths. 

6.  Twelve  and  seven  hundred  thirty-three  thou- 

sandths. 


DECIMAL  FRACTIONS  193 

7.  Fifty  and  one  hundred  seven  ten-thousandths. 

8.  Forty-eight  and  fifty-five  hundred-thousandths. 

9.  Eighty-four  and  nine  millionths. 

10.  537  hundred-thousandths.     47  millionths. 

11.  840  ten-thousandths.     435  thousandths. 

12.  507  millionths.     480  ten-thousandths. 

13.  46  hundred- thousandths.     420  thousandths. 

14.  326  ten-millionths.     25  billionths. 

15.  27  hundredths.     11  thousandths.     6  ten-thou- 

sandths. 

16.  3  millionths.     4  ten-thousandths.     5  ten-mil- 

lionths. 

17.  Forty-five  and  two  hundred  seventy-five  thou- 

sandths. 

18.  Six  and  twenty-five  ten-thousandths. 

19.  21,875  hundred-thousandths. 

20.  One  and  one  thousand  one-millionths. 

21.  Two  hundred  thirty-one  millionths. 

22.  2051  and  42  hundredths. 

23.  3  and  14  hundred  16  ten-thousandths. 
6.  Change  to  the  decimal  form  : 

275  1O      QQ/f      4  2  OO 

•  ToV-  1^-  ^y  MO"  "O<FO~-  L*' 

13    265     ft  5  23. 


4.  flft.          14.  341^. 

5.  rt^.        15.  527TM¥.  25. 

6.  ^          16. 


17      1 QQQ        365  O7 

J.  <  .      -LOtftfyo  OdOOO' 

8.     44i   m        i8.  247^7^.  28. 


94  9  &  1  Q      1  QAH      376.  9Q 

*     1000000'       lt7'     ±J7UIJ100000* 


4  OH      1  QOQ        231  Q.A 

lOOOOTT*       ^u*   loyyioooo0o* 
13 


194  ELEMENTARY  ARITHMETIC 

REDUCTION   OP   DECIMALS. 

Unlike  to  Like  Decimals. 
INDUCTIVE  STEPS. 

1.  ^  of  a  dollar  =  how  many  cents? 

1.  ^Lfo  of  a  dollar  =  how  many  cents? 

2.  Doe8^  =  $rWF? 

2.  What  is  the  difference,  then,  between  .7  and  .70  ? 

3.  What  effect,  then,  has  annexing  a  cipher  to  a  decimal  ? 


PRINCIPLE. 

Annexing  a  cipher  to  a  decimal  does  not  alter 
its  value. 


EXERCISES. 

1.  Eeduce  .5,  .47,  and  .046  to  like  fractions. 
Process.  Explanation. 

K  _       KAQ  *•  Thousandths  is  the  lowest  denomination. 

2.  We  must  reduce  .5  and  .47  to  thousandths. 

3.  Annexing  ciphers,  .6  =  .500,  and  .47  =  .470. 
.046  =•  .046               Principle :  Annexing  ciphers  to  a  decimal  does  not 

alter  its  value. 

RULE. 

Give  all  the  decimals  the  same  number  of  figures  by 
annexing  ciphers. 

2.  Reduce  the  following  to  like  decimal  fractions : 

1.  .69,  .034,  .0576.  6.  3.9,  5.24,  .34056. 

2.  .4,  .0536,  .00576.  7.  9.2,  1.146,  86.1136. 

3.  .06,  .005,  .005742.  8.  .72,  31.57,  .52405. 

4.  .004,  .053,  .00456.  9.  .004,  4.05,  4.0057. 

5.  4.6,  .573,  43.0568.  10.  9.2,  24,  .057246. 


REDUCTION  OF  DECIMALS  195 

3.  Reduce  the  following  to  like  decimals : 

1.  .08,  .75,  .006,  3.079. 

2.  .000135,  1.4067,  13.025. 

3.  63,  71.455,  315.7005,  6.15. 

4.  .409,  3.61,  75,  .10055,  19.6. 


A  Decimal  to  a  Common  Fraction. 
EXERCISES. 

1.  Reduce  .86  to  its  equivalent  common  fraction. 

Process.  Explanation. 

1.  Two  decimal  places  express  hundredths. 
•86  =  -fft  =  U-  2-  Therefore,  .86  -  rffr. 

3.  dfr  reduced  =  $£ 

RULE. 

1.  Omit  the  decimal  point. 

2.  Write  the  denominator. 

3.  Reduce  the  fraction  to  its  lowest  terms. 

2.  Reduce  to  common  fractions  : 


1. 

.36. 

11. 

.0625. 

21. 

.0075. 

31. 

.058. 

2. 

.75. 

12. 

.0375. 

22. 

.112. 

32. 

.0725. 

3. 

.45. 

13. 

.0750. 

23. 

.405. 

33. 

.0065. 

4. 

.50. 

14. 

.0500. 

24. 

.0032. 

34. 

.0562. 

5. 

.95. 

15. 

.0875. 

25. 

.0048. 

35. 

.5064. 

6. 

7. 
8. 

.500. 
.375. 
.625. 

16. 
17. 
18. 

.00500. 
.05625. 
.47043. 

26. 

27. 
28. 

.3525. 
.0108. 
.0002. 

36. 
37. 

38. 

.58*. 
.83*. 
.008f. 

9.  .875.       19.  .270496.       29.  .0006.        39.  .583J. 
10.  .125.       20.  .000047.       30.  .0014.        40.  .003f. 


196  ELEMENTARY  ARITHMETIC 

A  Common  Fraction  to  a  Decimal. 
INDUCTIVE   STEPS. 

1.  1  =  how  many  tenths  ? 
„    2  =  how  many  tenths  ? 

%  of  2,  or  f ,  =  how  many  tenths  ? 

2.  If  we  annex  0  to  2,  making  the  fraction  -^-,  and 
point  off  one  decimal  place  in  the  quotient,  what  will  be 
the  result  ? 

^L  =  how  many  hundredths  ? 

3.  If  we  annex  00  to  4,  making  the  fraction  ^jk,  and 
point  off  two  decimal  places  in  the  quotient,  will  not  the 
result  be  the  same,  .08  ? 


PRINCIPLE. 

A  decimal  place  must  be  cut  off  in  the  quotient 
for  every  cipher  annexed  to  the  numerator. 


EXERCISES. 

1.  Reduce  f  to  an  equivalent  decimal. 
Process.  Explanation. 

JkJLQJL  —  .375.  1.  To  render  the  division  exact  we  annex  three 

decimal  ciphers. 

2.  Dividing  3000  thousandths  by  8,  we  obtain  375  thousandths. 

3.  Pointing  off  three  decimal  places,  we  have  .375. 

2.  Reduce  f  to  a  decimal. 

Process.  Explanation. 

6-0  0    .85£.         !•  ^  wiU  n°t  exactly  divide  a  number  ending  in  0. 

2.  We  must,  however,  annex  ciphers. 

3.  600  hundredths  divided  by  7  —  85f  hundredths. 

4.  Pointing  off  two  decimal  places,  we  have  .85f. 

In  many  cases  the  common  fraction  may  be  omitted  as  unimportant. 


ADDITION  OF  DECIMALS  197 

3.  Reduce  to  decimals  : 

1.  A.        8.  «.       15.  fp      22.  fa  29. 

2.  if        9.  ^.     16.  Jp      23.  ftf.  30. 

3.  |.        10.  if.       17.  If-      24.  |f  31.  12&. 

4.  |.        11.  ^.       18.  A.      25.  f  32.  17A. 


6.  ft.      13.  T^.     20.  ft.      27.  jift.    34. 

7.  A.      14.  ^     21.  ft.      28.  JftL.    35. 


ADDITION  OP  DECIMALS. 

Since  decimals  and  integers  belong  alike  to  the  decimal 
system,  the  process  of  adding  decimals  does  not  essen- 
tially differ  from  that  of  adding  integers,  which  requires 
the  numbers  and  orders  to  be  added  to  be  of  like  name. 


PRINCIPLE. 

If  the   decimals  to   be  added   are  unlike,   they 
must  be  reduced  to  like  decimals. 


EXERCISES. 
1.  Find  the  sum  of  2.47,  4.364,  .0564. 

Process.  Explanation. 

2.47        =  2.4700  !•  The  given  fractions  are  unlike. 

4  oc 4     A  ^f)40  2-  They  must  be  reduced  to  like  fractions. 

3.  The  lowest   given  denomination  is  ten- 

.Uoo4  =     .05o4          A,  ,A, 

: thousandths. 


6.8904        6.8904  4.  Annexing   ciphers,    .47  becomes   .4700; 

.364  becomes  .3640. 
5.  Writing  like  orders  in  the  same  column  and  adding,  we  have  6.8904. 


198  ELEMENTARY  ARITHMETIC 

The  process  shows  that  the  decimal  points  of  the  given  numbers,  and 
that  of  their  sum,  stand  in  the  same  vertical  line,  and  that  in  practice  the 
ciphers  required  by  reduction  may  be  omitted. 

2.  Find  the  sum  of: 

1.  4.36,  .537,  49.52.  8.  3.054,  42.307,  .0006. 

2.  2.7,  43.54,  .0546.  9.  47.5,  2.736,  42.439. 

3.  46,  3.486,  2.057.  10.  495.3,  2.604,  5.3976. 

4.  3.2,  4.394,  57.3.  11.  4.670,  379,  42.574. 

5.  .4679,  33.10,  .536.  12.  46.74,  37.9,  357.60. 

6.  16.39,  25.46,  32.84.  13.  .3295,  32.95,  329.5. 

7.  46.38,  .2375,  29.54.  14.  37.54,  27.986,  38.45. 

3.  Find  the  sum  of: 

1.  30.062,  57.6203,  5620.07. 

2.  105.7,  5.0027,  29.9,  947.13. 

3.  400.07,  27.4,  987.09,  4.019,  470.9. 

4.  4.07,  39.0625,  900.07,  36.065,  219.107. 

5.  23.873,  .5625,  678.9,  19,719,  56.81. 

6.  625.25,  20.029,  3075.33,  927.8,  729.006. 

7.  7.29,  39.3039,  809.14,  90.075,  71.5. 

8.  301.5,  7.512,  6140.11,  114.3,  9.763. 

9.  27.725,  .6833,  9080.09,  78,006,  214.72. 

10.  151.39,  19.058,  1900.07,  6.705,  80.8. 

11.  23.04,  8.6796,  .0005,  7.00019, 


PROBLEMS. 

1.  What  is  the  sum  of  four  and  47  hundredths,  five 
and  758  thousandths,  twenty-five  and  475  thousandths  ? 

2.  What  is  the  sum  of  897  and  9  ten-thousandths,  17 
millionths,  18  thousandths,  98  ten-millionths,  167  hun- 
dred-thousandths, and  195  ten-millionths. 

3.  A  merchant's  sales  were  as  follows  :    On  Monday, 


I 

SUBTRACTION  OF  DECIMALS  199 

$470.45;  on  Tuesday,  $307.29 ;  on  Wednesday,  $584.40 ; 
on  Thursday,  $579.48;  on  Friday,  $225.36;  on  Saturday, 
$617.21.  Find  the  total  amount  of  his  sales. 

4.  52^,  240,  34-^  12^Wo>  3^^- 

5.  A  farmer  sold  at  different  times  the  following  quan- 
tities of  hay:    3.75   tons,   14.165   tons,  375.16247  tons, 
54.8125  tons,  18.5  tons,  21.75  tons,  and  25  tons.     How 
many  tons  did  he  sell  ? 


SUBTRACTION  OP  DECIMALS. 


PRINCIPLE. 

If   the   decimals   to   be   subtracted   are   unlike, 
they  must  be  reduced  to  like  decimals. 


EXERCISES. 
1.  From  35.34  subtract  9.6735. 


Process.  Explanation. 

5  34       —  35.3400  *•  The  given  decimals  are  unlike. 

q  £795  —  *    Q  67^5  ^'  ^hey  must  be  reduced   to   like  deci- 

inals. 


25.6665         25.6665  3    The  lowest  denomination  is  10,000ths. 

4.  Annexing  ciphers,  .34  becomes  .3400. 

5.  Writing  like  orders  in  the  same  column,  and  subtracting,  we  have 
25.6665. 

NOTE.  —  Be  careful  to  place  a  point  in  the  remainder  directly  under  the 
points  in  the  given  numbers. 


;>• 
200  ELEMENTARY  ARITHMETIC 

2.  What  is  the  difference  between : 

1.  38.46  and  26.53?  11.  5.94  and  .5947? 

2.  26.53  and  14.575  ?  12.  4.39  and  .0547  ? 

3.  47.49  and  32.576?  13.  .76  and  .076? 

4.  94.43  and  77.486  ?  14.  .0294  and  .001426  ? 

5.  52.97  and  33.40?  15.  .108  and  .0456? 

6.  47.53  and  24.355?  16.  400.07  and  27.4? 

7.  38.29  and  7.5467?  17.  900.07  and  36.065? 

8.  42.3  and  22.654?  18.  301.5  and  7.512? 

9.  .37  and  5.683?  19.  6140.11  and  114.3? 
10.  7.386  and  4.3956  ?  20.  27.725  and  .6833? 

3.  Find  the  value  of: 

1.  3.46  —  .075  +  4.34  —  2.3466. 

2.  4.683  +  3.47  —  .526  —  3.7243. 

3.  6.24  +  .430  —  5.275  —  .00056. 

4.  5.7  +  3.4607  —  2.005  —  4.4. 

5.  3.8004  —  1.00005  +  4.8  —  5.0506. 

6.  400.07  —  27.4  +  987.09  —  4.019  +  470.9. 

7.  7.29  +  39.3039  +  809.14  —  90.075  —  71.5. 

8.  301.5  —  7.512  -f  6140.11  —  114.3  —  9.763. 

9.  27.725  —  .6833  +  9080.09  —  78.006  —  214.72. 
10.  151.39  +  19.058  -f  1900.07  —  6.705  —  80.8. 

PROBLEMS. 

• 

1.  From  a  piece  of  cloth  containing  67.35  yards,  24f 
yards  were  cut.     How  many  yards  remained  ? 

2.  A  metre  is  39.3704  inches.     A  seconds-pendulum  is 
39.1392  inches  in  length.    Find  the  difference  of  length. 

3.  Two  men  walk,  respectively,  26.7  miles  and  22.94 
miles  per  day.     How  much  farther  does  the  first  walk 
than  the  second  ? 


MULTIPLICATION  OF  DECIMALS  201 

4.  Find  the  value  of  $385.75  —  $197.89. 

5.  Find  the  value  of  84  X  1.13  —  (66  —  1.2  X  2.4)  + 
100  X  (4  X  .018  +  .189). 

6.  A  clerk  has  a  yearly  salary  of  $1000.     He  pays 
$312  for  board,  $157.50  for  clothing,  and  $372.25  for  all 
other  expenses.     How  much  does  he  save  in  a  year  ? 

7.  A  man  who  owed  $699.60,  paid  $164.87.      How 
much  did  he  still  owe  ? 

8.  Find  the  sum  and  difference  of  .79864  and  .801. 

9.  A  merchant  owned   64.803   acres   of   land.      He 
bought  10.7045  acres,  and  sold  all  but  16.455  acres.    How 
many  acres  did  he  sell  ? 

10.  A  lady's  dress  cost  $13|,  her  bonnet  $5J,  her  shoes 
$2f ,  her  fan  $J.  Twenty-five  dollars  were  presented  in 
payment.  How  much  change  was  received  ? 


MULTIPLICATION    OP   DECIMALS. 

INDUCTIVE  STEPS. 

L  TO-  X  TO-  =  what?     -1  X  -1  =  what? 
2.  Both  factors  have  how  many  decimal  places?    Their 
product  has  how  many  decimal  places  ? 

3-  TO  X  TOT  =  what?     -1  x  -01  =  what? 

4.  Both  factors  have  how  many  decimal  places  ?    Their 

product  has  how  many  decimal  places  ? 


PRINCIPLE. 

The  product  of  two  decimals  contains  as  many 
decimal  places  as  both  the  decimals. 


202 


ELEMENTARY  ARITHMETIC 


EXERCISES. 
1.  What  is  the  product  of  .426  and  .34? 


Process. 
.426 
.34 
1704 
1278 
.14484 


2.  Multiply  .125  by  .06. 


Explanation. 

1.  426  X  34  =  14484. 

2.  Both  factors  have  3  -f-  2,  or  5  decimal  places. 

3.  We  must  point  off  5  decimal  places  in  the  product  ? 

4.  .426  X  .34  =  .14484.     [Principle.] 

Or,  we  may  say,  «  .426  =  tffo  and  .34  ==  rffr ;  ^ftfr  X 
== -14484." 


Process. 
.125 

.06 
.00750 


Explanation. 

1.  125  X  6  =  750. 

2.  5  decimal  places  are  required. 

3.  Prefixing  ciphers  we  have  .00750. 


3.  Multiply: 

"  1.  .43  by  .48. 

2.  .53  by  3.5. 

3.  .67  by  39. 

4.  .83  by  .406. 

5.  .93  by  .057. 

6.  1.027  by  .425. 

7.  .936  by  3.74. 

8.  .534  by  4.7. 

9.  32.8  by  .045. 

10.  4.34  by  .0067. 

11.  .270  by  4.053. 

12.  35.3  by  4.86. 

13.  27.9  by  .036. 

14.  .923  by  .0045. 

15.  4.57  by  .00537. 


16.  .3254  by  .0053. 

17.  .2704  by  .00476. 

18.  1.905  by  !o345. 

19.  20.27  by  .0057. 

20.  34.08  by  .00365. 

21.  34.05  by  4.2706. 

22.  406.03  by  2.5. 

23.  7.09  by  .0304. 

24.  30.701  by  .575. 

25.  937.06  by  .65. 

26.  9.704  by  40.7. 

27.  19.07  by  .16$. 

28.  11095  by  2.9. 

29.  3.097  by  .075. 

30.  .0035  by  .0005. 


MULTIPLICATION  OF  DECIMALS    •  203 

PROBLEMS. 

1.  What  will  300  bushels  of  wheat  cost  at  $1.25  per 
bushel  ? 

2.  A  merchant  sold  12.35  yards  of  silk  at  $3.15  per 
yard.     How  much  did  he  receive  for  it  ? 

3.  Find  the  cost  of  20.5  tons  of  hay  at  $12.375  a  ton. 

4.  Bought  14.75  yards  of  gingham  at  14  cents  a  yard. 
What  was  the  cost  of  the  piece  ? 

5.  What  is  the  cost  of  976  yards  of  cloth  at  $1.37| 
per  yard  ? 

6.  Find  the  cost  of  140  sacks  of  guano,  each  sack 
containing  162J  pounds,  at  $17f  a  ton. 

7.  What  is  the  value  of  1648  bushels  of  wheat  at 
$.621  per  bushel  ? 

8.  What  cost  250  yards  of  carpet  at  $1.60  per  yard? 

9.  What  cost  64  barrels  of  apples  at  $2.50  per  barrel  ? 

10.  What  cost  34.8  yards  of  cloth  at  87-1-  cents  per  yard  ? 

11.  What  cost  24  yards  of  silk  at  $1.16|  per  yard  ? 

12.  What  will  840  bushels  of  oats  cost  at  25  cents  a 
bushel  ? 

13.  What  will  26  loads  of  lime  cost,  each  containing 
15^  bushels,  at  22^  cents  a  bushel  ? 

14.  What  are  17.6  acres  +  23.25  acres  +  42.625  acres 
worth  at  $40  per  acre  ? 

15.  What  are  45  dozen  eggs  worth  at  $.12^-  per  dozen? 

16.  231  cubic  inches  =  one  gallon;  31.5  gallons  =  1 
barrel.     How  many  cubic  inches  in  a  barrel  ? 

17.  A  farmer  bought  .75  bushels  of  grass  seed  at  $5  a 
bushel.     Find  the  cost. 

18.  A  man  owning  .4236  of  a  yacht  sold  .3  of  his 
share.     What  part  had  he  left  ? 


204  *       ELEMENTARY  ARITHMETIC 

DIVISION  OF  DECIMALS. 

INDUCTIVE   STEPS. 

l-  A  X  fV  =  what?  .3  X  -4  =  what?  What  are 
.3  and  .4  called  in  relation  to  their  product  ?  Factor  X 
factor  =  what  ?  .3  X  -04  =  what  product  ?  How  many 
decimal  places  are  in  the  product  ? 

2.  The  product  must  always  he  given  as  many  decimal 
places  as  there  are  in  what  ? 

3.  The  dividend  is  the  product  of  what  two  factors  ? 

4.  The  dividend  contains  as  many  decimal  places  as 
both  the and  the . 

5.  If  the  dividend  has  6  decimal  places  and  the  divisor 
has  4  decimal  places,  how  many  decimal  places  must  the 
quotient  have  ? 


PRINCIPLE. 

The  quotient  contains  as  many  decimal  places 
as  the  number  of  decimal  places  in  the  dividend 
exceeds  the  number  in  the  divisor. 


EXERCISES. 
1.  Divide  .08128  hy  .32. 

Process.  Explanation. 

253  i.  8128  -r-  32  ==  254. 

-32). 08128  (.254  2.  .08128  has  5  decimal  places;  .32  has  2 

decimal  places. 

172  3.  The  quotient  must  have  5  —  2,  or  3  deci- 

160  mal  places. 

128  4.  Hence,  the  quotient  required  is  .254. 

128 

Proof. 

.254  X  -32  =  .08128. 


DIVISION  OF  DECIMALS  205 

2.  Divide  17.28  by  .00144. 

Process.  Explanation. 

55  oi.  The  divisor  .00144  has  5  decimal 

.00144  )  17.28000  (  12000        places  ;  the  dividend  cannot  have  a  less 

number. 

288  2.  Annexing  three  ciphers  to  17.28, 

288  we  have  17.28000. 

000  3.  5  —  5  =  0;  the  quotient,  there- 

fore, can  have  no  decimal  place. 
4.  Dividing,  we  have  for  quotient  the  integer  12,000. 

3.  Divide  7.3  by  3650. 

Process.  Explanation. 

033  i.  Annexing  two  ciphers  to  7.3,  we  have 

3650)  7.300  (.002         7.300. 

7.300  2.  3650  has  no  decimal  place ;  7.300  has  3 

decimal  places. 

3.  The  quotient  must  have  3  —  0,  or  3  decimal  places. 

4.  Hence,  the  quotient  required  is  .002,  obtained  by  prefixing  two 
ciphers. 

RULE. 

1.  Divide  •without  regard  to  the  decimal  point. 

2.  Should  the  dividend  lack  figures,  annex  ciphers. 

3.  After  dividing,  give  the  quotient  as  many  decimal 
places  as  the  number  of  decimal  places  in  the  dividend 
exceeds  those  in  the  divisor. 

4.  Should  the  quotient  lack  figures,  prefix  ciphers. 

4.  Divide: 

1.  2.176  by  .34.  6.  2.1824  by  .034. 

2.  .07245  by  .23.  7.  405.15  by  .111. 

3.  16.5  by  .25.  8.  45.625  by  .125. 

4.  .0864  by  .24.  9.  .0125  by  2.5. 

5.  3.024  by  .07.  10.  58.794  by  12.3. 


206 


ELEMENTARY  ARITHMETIC 


11.  .0043  by  .230. 

12.  .065  by  .50. 

13.  7  by  350. 

14.  1.2  by  3.60. 

15.  75  by  .025. 

16.  1.075  by  .43. 

17.  1.5652  by  .043. 

18.  3.024  by  .07. 

19.  739.44  by  .009. 

20.  185.175  by  .015. 

21.  10.24  by  320. 

22.  600  by  .625. 

23.  6.256  by  .375. 

24.  .03876  by  .19. 

25.  40  by  640. 

26.  7.6  by  .0304. 

27.  .18312  by  .056. 

28.  .12126  by  .235. 


29.  .0169  by  .013. 

30.  80.010  by  .009. 

31.  16.1262  by  3.06. 

32.  1.3621  by  .514. 

33.  .016074  by  .047. 

34.  1.25  by  .015. 

35.  65  by  .0039. 

36.  .0402  by  150. 

37.  3647  by  .125. 

38.  72  by  .064. 

39.  16.02  by  .045. 

40.  34.4088  by  1.62. 

41.  .291624  by  2.32. 

42.  30,000  by  .000003. 

43.  102.102  by  102. 

44.  1.1502  by  .0027. 

45.  .0342568  by  .006523. 

46.  .987650  by  .0000125. 


The  Decimal  Point  as  Multiplier  and  Divisor. 
INDUCTIVE   STEPS. 

1.  .001  =  what?    0.01  =  what?    00.1  =  what?    What 
has  been  done  with  the  decimal  point?     How  does  .01 
compare  in  value  with  .001  ?     How  does  .1  compare  in 
value  with  .01  ? 

2.  Moving  the  decimal  point  one  place  to  the  right  has 
what  effect  upon  a  number  ?    Moving  the  point  two  places 
has  what  effect  ?     Three  places  ?     Four  places  ? 

3.  .1  =  what  ?    .01  =  what  ?    .001  =  what  ?    What  has 
here  been  done  with  the  decimal  point  ?     How  does  .01 


DIVISION  OF  DECIMALS  207 

compare  in  value  with  .1  ?  How  does  .001  compare  in 
value  with  .01  ?  How  does  .001  compare  in  value  with  .1  ? 
4.  Moving  a  decimal  point  to  the  left  has  what  effect 
upon  a  number  ?  Moving  the  point  two  places  has  what 
effect  ?  Three  places  ?  Four  places  ?  Five  places  ? 


PRINCIPLES. 

1.  Every  removal  of  a  decimal  point  one  place 
to  the  right  multiplies  the  number  by  1O. 

2.  Every  removal  of  a  decimal  point  one  place 
to  the  left  divides  the  number  by  1O. 


EXERCISES. 

1.  Multiply  7.943  by  100. 

Process.  Explanation. 

794.3  1-  There  are  two  ciphers  in  the  multiplier. 

2.  We  therefore   move   the   point  two  places   towards    the 
right  and  have  794.3. 

2.  Multiply: 

1.  39.63  by  10.          6.  .95436  by  10,000. 

2.  49.306  by  100.      7.  .8  by  10;  by  100;  by  1000. 

NOTE. — If  there  are  not  enough  places,  annex  ciphers. 

3.  3.946  by  1000.      8.  .2000  by  1,000,000. 

4.  .495  by  100.          9.  .00013  by  100,000. 

5.  6.387  by  10.        10.  .3041  by  10,000. 

3.  Multiplication  by  a  decimal  has  the  effect  of  making 
the  product  less  than  the  multiplicand ;  hence,  to  multiply 
by  .1,  .01,  or  .001,  etc.,  we  move  the  point  toward  the 
left. 


208  ELEMENTARY  ARITHMETIC 

4.  Multiply: 

1.  39.63  by  .1.  3.  3.946  by  .001. 

2.  49.306  by  .01.  4.  .495 'by  .01. 

WRITTEN    EXERCISES. 

1.  Divide  436.58  by  100. 

Process.  Explanation. 

4.3658.  !•  There  are  two  ciphers  in  the  multiplier. 

2.  Therefore,  we  move  the  point  two  places  toward  the 
left,  and  have  4.3658. 

2.  Divide: 

1.  403.6  by  100.  6.  53.95  by  10,000. 

2.  3756  by  10.  7.  .8  by  10;  by  100;  by  1000. 

3.  470.6  by  1000.  8.  .00013  by  100,000. 

4.  4.825  by  100.  9.  2000  by  1,000,000. 

5.  38.62  by  1000.  10.  .3041  by  100,000. 

Division  by  a  decimal  has  the  effect  of  making  the 
quotient  larger  than  the  dividend;  hence,  to  divide  by  .1, 
.01,  or  .001,  etc.,  we  move  the  point  toward  the  right. 

3.  Divide: 

1.  403.6  by  .01.  3.  470.6  by  .001. 
403.6  --  .01  =  40360.            4.  756.9  by  .0001. 

2.  37.56  by  .1.  5.  4.825  by  .001. 

PROBLEMS. 

1.  I  bought  a  farm  containing  125  acres  for  $6843.75 
What  was  the  price  per  acre  ? 

2.  At   $3   a  yard,  how  many  yards  of  cloth  can   be 
bought  for  $546  ? 


DIVISION  OF  DECIMALS  209 

3.  If  4.7  acres  of  land  produce  131.6  bushels  of  wheat, 
what  is  the  average  crop  per  acre  ? 

4.  How  many  dozen  eggs,  at  $.12}  a  dozen,  can  be 
bought  for  $12? 

5.  How  many  yards  of  velvet,  at  $4  a  yard,  can  be 
bought  for  $23? 

6.  How  many  pounds  of  tea  can  be  bought  for  $6.75 
at  75  cents  a  pound  ? 

7.  At  $12.375  a  ton,  how  many  tons  of  hay  can  be 
bought  for  $2326.50  ? 

8.  How  many  tons  of  freight  at  $2.1 2 \  per  ton  can  be 
transported  for  $107.07? 

9.  At  $10.45  per  barrel,  how  many  barrels  of  flour 
can  be  bought  for  $1055.45? 

10.  At  $.31  \  a  bushel,  how  many  bushels  of  potatoes 
can  be  bought  for  $9  ? 

11.  How  many  dress  patterns  of  12.50  yards  each  can 
be  cut  from  4  pieces  of  French  muslin  containing  25  yards 
each? 

12.  Find  a  man's  daily  wages  when  he  was  paid  $27.70 
for  22  days'  work  ? 

13.  Paid  $40.50  for  a  pile  of  wood,  at  the  rate  of  $3.37  J 
a  cord.     How  many  cords  were  in  the  pile  ? 

14.  How  many  oranges,  at  8J  cents  each,  will  $1.50 
buy? 

15.  Find  the  price  of  each : 

1.  If  250  bushels  of  corn  cost  $125.00. 

2.  If  70  pounds  of  sugar  cost  $2.80. 

3.  If  288  bushels  of  wheat  cost  $259.20. 

4.  If  200  acres  of  land  cost  $4267}. 

5.  If  18  turkeys  cost  $15.75. 

14 


210  ELEMENTARY  ARITHMETIC 

6.  If  792  pounds  of  rice  cost  $39.60. 

7.  If  2000  pounds  of  butter  cost  $350. 

8.  If  2500  pounds  of  beef  cost  $156.25. 

9.  If  1234  pineapples  cost  $98.72. 
10.  If  4|  tons  of  coal  cost  $25.41J. 

16.  Find  the  value  of: 

1.  2.24  X  6  H-  28. 

2.  $30.10  -f-  10  X  1000. 

3.  8.2  X  9.3  —  (45f  --  12.5). 

4.  (25  X  .5  X  12  +  20)  --  100. 

5.  3.71  +  2.64  -v-  160  +  7.55  X  .07  +  .071  X  25. 

6.  (15  —  10  X  .3)  X  6.192  -=-  (7  X  5.4  —  35.048). 

7.  (4.625  +  1.146)  — (1.2  +  3.571). 

8.  1.5  X  .08  X  .5. 

9.  94.5  -T-  250  +  16  -r-  (4.5  -=-  .225)  +  87.25  -f- 

(1.6  —  .35). 
10.  (.48  -*-  800  X  10,000  +  6.4  —.08)  -h  .125. 

UNITED  STATES  MONEY. 

United  States  Money,  as  we  have  already  seen,  is  ex- 
pressed in  the  decimal  system.  Its  denominations  and 
their  relation  to  one  another  are  as  follows : 

Table. 

10  mills     make  1  cent  (c.). 
10  cents     make  1  dime  (d.). 
10  dimes    make  1  dollar  ($). 
10  dollars  make  1  eagle  (E.). 

The  dollar  is  the  unit. 
$5J  is  written  decimally  $5.50,  or  $5T6T$J-. 
$5  and  5  cents  is  written  decimally  $5.05,  or 
$5  and  5  mills  is  written  decimally  $5.005. 


UNITED  STATES  MONEY  211 

1.  Read  the  following : 

1.  $25.85.  6.  $22.36.  11.  $670.086. 

2.  $26.965.  7.  $210.210.  12.  $3056.002. 

3.  $35.355.  8.  $256.006.  13.  $6789.012. 

4.  $255.236.         9.  $300.003.  14.  3456.789. 

5.  $36.05.  10.  $505.505.  15.  $1798.365. 

2.  Write  the  following  in  figures  : 

1.  Seven  dollars,  twenty-five  cents. 

2.  Ten  dollars,  forty  cents,  five  mills. 

3.  Forty-five  dollars,  fifty-four  cents. 

4.  Sixty-five  dollars,  eight  cents. 

5.  Seven  eagles,  seven  dollars,  seven  cents. 

6.  Two  hundred  dollars,  six  and  a  half  cents. 

7.  Ninety  dollars,  eight  dimes,  8J  cents. 

8.  One  thousand  fifty-six  dollars,  94J  cents. 

9.  Eight  thousand  seventy-nine  dollars. 
10.  One  dollar,  one  cent,  one  mill. 

ORAL    EXERCISES. 

1.  $1  equals  how  many  cents  ?     $|  ?     $J  ?     $^  ? 

2.  $4  equal  how  many  cents  ? 

3.  J  cent  equals  how  many  mills  ?     £  cent  ? 

4.  $J-  equals  how  many  cents  ?     $|  ?     $  J  ? 

5.  $1  equals  how  many  mills  ? 

6.  20  mills  equal  how  many  cents  ?     40  mills  ? 

7.  30  cents  equal  how  many  dimes  ?     50  cents  ? 

8.  30  dimes  equal  how  many  dollars  ?     60  dimes  ? 

9.  200  cents  equal  how  many  dollars  ?     500  cents  ? 

10.  $20  equal  how  many  eagles  ?     $40  ? 

11.  10  cents  is  what  part  of  a  dollar?     20  cents?     50 
cents  ? 


212  ELEMENTARY   ARITHMETIC 

12.  25  cents  is  what  part  of  a  dollar?     30  cents?     75 
cents  ? 

The  following  rules  are  obvious  : 

1.  To  change: 

(1)  Cents  to  mills,  multiply  by  1O. 

(2)  Dollars  to  cents,  multiply  by  1OO. 

(3)  Dollars  to  mills,  multiply  by  1OOO. 

2.  To  change: 

(1)  Mills  to  cents,  divide  by  1O. 

(2)  .Cents  to  dollars,  divide  by  1OO. 

(3)  Mills  to  dollars,  divide  by  1OOO. 

NOTE. — The  proper  placing  of  the  decimal  point  in  a  given  numher 
will  effect  the  change  required. 

13.  Change: 

1.  12,345  mills  to  dollars. 

2.  Six  cents  and  six  mills  to  dollars. 

3.  57  cents  to  mills.          17.  2700  mills  to  dollars. 

4.  43  mills  to  cents.         18.  2956  mills  to  dollars, 

5.  46  cents  to  mills.         19.  3548  mills  to  dollars. 

6.  57  mills  to  cents.         20.  $70.30  to  cents. 

7.  47  dollars  to  cents.      21.  $27.35  to  cents. 

8.  35  dollars  to  cents.      22.  $29.03  to  cents. 

9.  296  dollars  to  cents.    23.  $56.38  to  cents. 

10.  326  dollars  to  cents.    24.  $7.866  to  mills. 

11.  4600  cents  to  dollars.  25.  5795  cents  to  dollars. 

12.  900  cents  to  dollars.    26.  6594  cents  to  dollars. 

13.  836  cents  to  dollars.    27.  7984  cents  to  dollars. 

14.  2548  cents  to  dollars.  28.  $200.002  to  mills. 

15.  26  dollars  to  mills.      29.  $404.404  to  mills. 

16.  9  dollars  to  mills.        30.  6  E.  and  $6  to  cents. 


UNITED  STATES  MONEY  213 

ORAL    PROBLEMS. 

1.  Jack  had  25   cents,  and  his  father  gave  him  50 
cents.     How  much  had  he  then  ? 

2.  Mr.  Duane  divided  $2.30  equally  among  5  girls. 
HOW  much  did  each  receive  ? 

3.  Out  of  $6.00,  a  man  spent  $2.50.     How  much  had 
he  left. 

4.  A  skilled  workman  earned  $3.50  per  day.     How 
much  did  he  earn  in  6  days  ? 

5.  A  lad  received  for  services  rendered  55  cents.     He 
spent  10  cents  for  a  toy,  20  cents  for  a  novel,  and  15  cents 
for  pens  and  ink.     How  much  had  he  left  ? 

6.  How  much  will  5  tons  of  coal  cost  at  $5.50  per  ton  ? 

7.  I  have  one  dollar  to  spend.     I  pay  J  dollar  for  a 
book,  and  J  dollar  for  pens,  ink,  and  paper.     How  much 
remains  ? 

8.  If  a  gentleman  buys  a  barrel  of  flour  for  six  dol- 
lars and  fifty  cents,  and  hands  the  seller  a  ten-dollar  bill, 
how  much  change  should  he  receive  ? 

9.  A  man  bought  a  horse  for  $150.     He  kept  it  at  an 
expense  of  $60.    He  then  sold  it  for  $225.25.    How  much 
did  he  gain  ? 

10.  At  the  rate  of  $7.50  a  barrel,  what  will  £  of  a  bar- 
rel of  flour  cost? 

11.  What  would  be  the  cost  of  10  yards  of  cloth  at 
$2.75  per  yard  ? 

12.  A  fruit-vender  sold  6  apples  at  3  cents  apiece,  and 
8  oranges  at  5  cents  apiece.     If  they  cost  ^5-  of  a  dollar, 
how  much  did  he  gain  ? 

13.  Find  the  cost  of  3  dozen  copy-books,  at  $1.10  per 
dozen. 


214  ELEMENTARY  ARITHMETIC 

14.  Henry  had  f  of  a  dollar  and  spent  £  of  a  dollar. 
How  many  cents  had  he  remaining  ? 

15.  George  has  $2.625  and  Henry  has  $3.375.     If  they 
share  their  money  equally,  how  many  dollars  has  each  ? 

•WRITTEN   PROBLEMS. 

1.  Jack  gives  $1.62J  for  a  pair  of  shoes,  37  J  cents  for 
a  penknife,  and  25  cents  for  a  baseball.     How  much  does 
he  pay  for  all  ? 

2.  A  man  is  indebted  to  A.,  $740.59 ;  to  B.,  $36 ;  to 
C.,  $.985 ;  to  D.,  $1.04.     How  much  does  he  owe  ? 

3.  Find  the  sum  of  19  dollars,  7  cents,  5  mills;    20 
dollars,  9  cents,  9  mills ;  24  dollars,  23  cents,  6  mills. 

4.  Mr.  Rex  paid  for  repairs  as  follows :    Carpenter- 
work,  $424.30;    plastering,  $170.48;    plumbing,  $75.97; 
incidental  expenses,  $205.49.     How  much  did  he  pay  for 
repairs  ? 

5.  How  much  must  be  added  to  $70.039  to  make  the 
sum  $1106.39  ? 

6.  A  man  sold  54.6  acres  of  land,  which   cost  him 
$49.60  per  acre,  for  $3000.     How  much  did  he  gain  ? 

7.  How  much  must  you  add  to  $40.173  to  make  $100  ? 

8.  A  farmer  sold  52.375  pounds  of  butter  for  $7.856i. 
How  much  did  he  receive  a  pound  for  it? 

9.  If  one  pound  of  butter  costs  12J  cents,  what  will 
4  firkins  cost,  each  weighing  5#  pounds  ?. 

10.  If  5.3  yards  of  cloth  cost  $9.275,  what  will  8.5  yards 
cost? 

11.  Bought  a  roll  of  carpet,  containing  82  yards,  for 
$45,  and  sold  it  for  75  cents  a  yard.     How  much  did  I 
gain? 


UNITED  STATES  MONEY  215 

12.  A  shoemaker  sells  35  pairs  of  shoes  for  $70.35,  of 
which  21  pairs  are  sold  at  $2.25  a  pair.     At  what  price 
per  pair  are  the  rest  of  the  shoes  sold  ? 

13.  A  man  worked  9  days  for  $2.1 2 \  per  day.     How 
much  did  he  earn  ? 

14.  What  is  the  value  of  67.75  acres  of  land  at  $62.50 
per  acre  ? 

15.  When  tea  is  $.50  per  pound,  how  much  can  be 
bought  for  $.75  ? 

16.  At  $1.25  per  yard,  how  many  yards  of  cloth  can  be 
bought  for  $35  ? 

17.  How  many  pounds  of  butter,  at  33  J  cents  a  pound, 
can  be  bought  for  $32  ? 

18.  A  pound  sterling  is  worth  $4.8665.     What  are  38.8 
pounds  sterling  worth  ? 

19.  An  errand  boy  earns  $2.75  a  week.     In  how  many 
weeks  will  he  earn  $49.50  ? 

20.  A  man  earns  $12  a  week  and  spends  on  an  aver- 
age, $8.50  a  week.    In  how  many  weeks  will  he  save  $140. 

21.  How  many  quarts  of  berries,  at  $.08  a  quart,  will 
it  take  to  pay  for  4  yards  of  cloth  at  $.84  ? 

22.  A  merchant  sold  25.5  yards  of  cambric  at  $.20  per 
yard  and  gained  $1.70<     How  much  did  it  cost  him  ? 

23.  If  13.5  yards  of  cloth  cost  $84|,  what  will  23.75 
yards  cost? 

24.  If  I  earn  $70  per  month  and  spend  $45.50  of  it,  in 
how  many  months  will  I  save  $1080? 

25.  If  3  barrels  of  apples  cost  $19.125,  find  the  cost  of 
100  barrels  ? 

26.  If  a  man  spends  $.87  in  one  day,  how  much  will 
he  spend  in  15.5  days? 


216  ELEMENTAEY  ARITHMETIC 

27.  A  grocer  bought  10  barrels  of  sugar,  each  contain- 
ing 235  pounds,  for  $152.75.     How  much  did  it  cost  per 
pound  ? 

28.  Paid  $24  for  cuffs  at  16f  cents  per  pair.    How  many 
dozen  pairs  were  bought  ? 

29.  Bought  8  firkins  of  butter  for  $72,  and  gave  6  of 
them  for  7  yards  of  cloth.    What  was  a  yard  of  the  cloth 
worth  ? 

30.  A  load  of  hay,  at  75  cents  per  100  pounds,  cost 
$13.98.     "What  was  the  weight  of  the  hay  ? 

31.  When  -f  of  $785  was  spent,  how  much  remained? 

32.  If  a  workman  saves  $62.40  in  a  year  by  laying  up 
20  cents  each  day,  how  long  would  it  take  4  men  at  the 
same  rate  to  save  $124.80  ? 

33.  If  44|  yards  of  cloth  cost  $199,  how  much  must 
be  paid  for  80  yards  ? 

34.  An  income  of  4325  dollars  is  spent  as  follows :  J  at 
home,  and  J  of  the  remainder  abroad.     How  much  was 
spent  abroad? 


BILLS  AND  ACCOUNTS. 

DEFINITIONS. 

1.  A  Bill  is  a  written  statement  showing  the  quantity 
and  price  of  the  items  bought,  together  with  the  total 
cost. 

2.  A  bill  is  Receipted  when  the  words  "Bec'd  pay- 
ment," written  at  the  bottom,  are  followed  by  the  signa- 
ture of  the  maker  of  the  bill. 


BILLS  AND   ACCOUNTS 


217 


3.  The  person  who  owes  the  bill  is  called  the  Debtor. 

4.  The  person  to  whom  the  bill  is  owing  is  called  the 
Creditor. 

5.  The  Footing  of  a  bill  is  the  total  amount  of  it. 

6.  An  Account  is  a  bill  which  contains  both  debit  and 
credit  items. 

7.  The  Balance  of  an  Account  is  the  difference  be- 
tween the  amounts  of  the  debits  and  credits. 


Abbreviations  in  Common  Use. 


@,  at. 

Do.,  the  same. 

Mo.,  month. 

%,  account. 

Doz.,  dozen. 

No.,  or  $,  number. 

Acc't,  account. 

Dr.,  debtor. 

Pay't,  payment. 

Bal.,  balance. 

Fr't,  freight. 

Pd.,  paid. 

Bbl.,  barrel. 

Hhd.,  hogshead. 

Per,  by. 

Bo't,  bought. 

Inst.,  this  month. 

Rec'd,  received. 

Co.,  company. 

Int.,  interest. 

Tilt.,  last  month. 

Cr.,  creditor. 

Lb.,  pound. 

Yd.,  yard. 

Cts.,  cents. 


Mdse.,  merchandise.     Yr.,  year. 


PROBLEMS. 
1.  A  man's  account  at  a  store  stands  thus  : 


Dr. 

$4.745 

2.62J 

1.27 
.45 

5.28J 
10.25 

What  is  due  the  merchant  ? 


Cr. 


1.245 
•62J 
3.45 
1.87J 
5.25 


218 


ELEMENTARY  ARITHMETIC 


Copy,  fill  out,  and  find  the  footings  of  the  following 

2. 

BALTIMORE,  March  1,  1898. 
MR.  J.  B.  MOORE, 

Bought  of  WEBB  &  BOND  : 


75J  yards  of  Carpeting 
37    yards  of  Drugget 
8    Eugs 
5    Mats 
18    yards  Oil-Cloth 

@  $2.12J 
"     1.20 
"     4.16 
"     2.37J 
"     1.08 

$ 

$ 

3. 


MR.  JAMES  JOHNSON, 


ORLEANS,  May  20,  1898. 
To  LELONG  &  NOTT,  Dr. 


To  37  bbl.  Pork  @  $24.35 

"  127  bbl.  Flour  «       8.15 

"  3  hhd.  Molasses—  169  gal.  "  .43 
"  29  firkins  Butter—  2120  Ib.  "  .31 
"  3  boxes  Eaisins  "  4.65 


4. 

MR.  DAVID  DIXON, 


YORK,  April  1,  1898. 
To  SCHENCK  &  VAIL,  Dr. 


1882. 

Jan. 

9 

To  3  Gold  Watches—  $124.50 

,  $61.24, 

$57.18 

$ 

u 

Feb. 

13 
3 

"  437  pwt.  Gold  Chain 
"  35  sets  Plated  Tea-  Service 

@    1.15 
«  43.10 

a 

15 

u       7    u            u                   u 

"  51. 

\ 

$ 

BILLS  AND  ACCOUNTS  219 

5. 

BOSTON,  Jan.  1,  1870. 
DANIEL  CHAPMAN  &  Co., 

Bo't  of  PALMER  &  BROTHER  : 


67  pairs  Calf  Boots, 

@     $3.75 

$ 

108     "      Thick  " 

"        2.62 

75     "      Gaiters, 

"        1.12 

27     "      Buskins, 

"          .86 

35     "      Slippers, 

.70 

50     "     Rubbers, 

"        1.04 

$717 

93. 

Ree'd  Payment, 

PALMER  &  BROTHER, 

By  GEO.  BAKER. 

6.  Nov.  23,  1899.      Sold  to  W.  P.  Beaux  for  cash :  1 
No.  1  B.  W.  bedstead,  $22.00;  1  dressing  case  (18  x  36 
mirror),  $40.00 ;  excelsior,  .40 ;   glass  case,  .50 ;    marble 
box,  .75 ;  4  mats,  16  yds.  @  .12. 

Make  out  the  above  in  bill  form,  and  write  your  own  name  as  receiver 
of  the  cash. 

7.  Mrs.  B.  K.  Lex  bought  of  Rex  &  Brooke  15  yds. 
black  china  silk,  @  $.65 ;  7  yds.  green  henrietta  cloth,  @ 
$.50 ;  6  yds.  navy  blue  serge,  @  $.80 ;  2  felt  hats,  @  $.67 ; 

2  English  velour  capes,  @  $14.75. 

8.  Sold  Archibald  Weaver  24  sets  If  in.  No.  2  bed- 
casters,  @  13c. ;  40  sets  2  in.  ISTo.  1  bed-casters,  @  17c. ; 
24  sets  2  in.  No.  2  bedpost-casters,  @  23c. ;  18  sets  No. 

3  plate-casters,  @  lie.;   18  sets  I  H.  P.  W.  casters,  @ 
13c. ;  6  sets  brass  H.  &  W.  casters,  @  35c. 

9.  Mr.  David   Mason  bought   of  George   Lelong  & 
Brother  5  blank-books,  @  $2.30;    7  gross   Spencerian 


220  ELEMENTARY  ARITHMETIC 

pens,  @  $1.12};  15  B.  and  S.  book-keeping,  @  $1.75;  4 
reams  cap  paper,  @  $3.40;  20  Townsend's  commercial 
law,  @  $2.87} ;  12  packs  plain  cards,  @  37}c. ;  note-paper 
and  ink,  $2.78. 

10.  Mrs.  A.  M.  Williams  bought  of  Andrew  Jenkins 
&  Son,  37  chests  of  green  tea,  @  $23.75;  42  chests  of 
black  tea,  @  $17.50;  43  casks  of  wine,  @  $99;  12  crates 
of  Liverpool  ware,  @  $75 ;  19  barrels  of  Genesee  flour, 
@  $7.00 ;  23  bushels  of  rye,  @  60c. 

11.  Sold  Mrs.  Susan   Crockett  1  bbl.  sugar,  245  lb., 
@  3}c.;  10  lb.  oatmeal,  @  2}c.;  3  lb.  honey,  @  12}c.; 
4  sacks  flour,  @  $1.35;  3  lb.  raisins,  @  13c. ;  7  doz.  eggs, 
@  15c. ;  10  lb.  crackers,  @  8}c. ;  1  caddy  Japan  tea,  @ 
65c. ;  10  lb.  salt,  @  3}c. 


DENOMINATE    NUMBERS. 

DEFINITIONS. 

1.  A  Denominate  Number  is  a  concrete  number  whose 
unit  is  applied  to  measurement. 

3  feet,  8  quarts,  7  days,  are  denominate  numbers. 

2.  A   Simple   Denominate    Number   is   composed   of 
units  of  the  same  denomination. 

5  pints,  27  cubic  feet,  are  simple  denominate  numbers. 

3.  A  Compound  Denominate  Number  is  composed  of 
two  or  more  denominations  that  have  an  established  rela- 
tion to  each  other. 

4  feet,  6  inches  ;  3  bushels,  2  pecks,  1  quart,  are  compound  denominate 
numbers. 


DENOMINATE  NUMBERS 


221 


LINEAR    MEASURE. 
Linear  Measure  is  used  in  measuring  length. 

Table. 


12  inches  (in.)  =  1  foot  (ft.). 
3  feet  =  1  yard  (yd.). 

5£  yards 
16£  feet 
320  rods 


=  1  rod  (rd.). 
=  1  mile  (mi.). 


ORAL   EXERCISES. 

1.  How  many  inches  in  : 

1.3ft?     4ft.?     5ft.?     6ft?     7ft?     8ft.? 

2.  1  ft  ?     |  ft.  ?     }  ft  ?     |  ft  ?     |  ft  ?     f  ft  ? 

3.  1  yd.  ?     3  yd.  ?     4  yd.  ?     4|  yd.  ?     5  yd.  9  in.  ? 

2.  How  many  feet  in  : 

1.  3yd.?   4yd.?   5yd.?   10yd.?   25yd.?  112yd.? 

2.  2  rd.  ?    3  rd.  ?    4  rd.  ?    5  rd.  ?    10  rd.  ?    20  rd.  ? 

3.  24  in.  ?     36  in.  ?     60  in.  ?     132  in.  ?     144  in.  ? 

1728  in.  ? 

4.  J  yd.  ?      f  yd.  ?      f  yd.  ?      2J  yd.  ?      3f  yd.  ? 

5fyd.? 

3.  How  many  yards  in  : 

1.  4rd.?   8rd.?    10  rd.?    12  rd.?   20  rd.  ?   25  rd.  ? 

2.  12ft.?    18ft.?    24ft?   36ft?    144ft?   5280ft.? 

3.  16J  ft.?    lrd-?   3ft?    12  in.?   36  in.  ?    108  in.  ? 

4.  How  many  rods  in  : 

1.  Imi.  ?   2  mi.?   3  mi.  ?   8  mi.?    10  mi.  ?    12  mi.  ? 

2.  161  ft-?     51  Jd-?     33  ft.?     66  ft?     99  ft? 

22  yd.  ? 


222  ELEMENTARY  ARITHMETIC 

5.  How  many  miles  in  320  rd.  ?     640  rd.  ?     960  rd.  ? 
1280rd.?     1600  rd.? 

6.  5J  yd.  =  1  rod.    Compute  the  number  of  yd.  in  a  mi. 

7.  16 J  ft.  =  1  rod.    Compute  the  number  of  ft.  in  a  mi. 

SURFACE    MEASURE. 

1.  Surface  or  Square  Measure  is  used  in  measuring 
surface.  A  surface  has  only  length  and 
breadth. 


This  page  at  which  you  are  looking  is  a  surface. 


Angle. 

2.  An  Angle  is  the  difference  in  direc- 
tion of  two  lines  drawn  from  the  same 
point. 

3.  A  Square  has  four  equal  sides  and 


Square.  four  equal  angles.     The  equal  angles  are 

called  right  angles. 


Table. 


144  Square  Inches  (sq.  in.)  =  1  Square  Foot  (sq.  ft.). 

9  Square  Feet  =  1  Square  Yard  (sq.  yd.). 

30£  Square  Yards  =  1  Square  Rod  (sq.  rd.). 

160  Square  Rods  =  1  Acre  (A.). 

640  Acres  =  1  Square  Mile  (sq.  mi.). 


ORAL   EXERCISES. 
1.  How  many  square  inches  in : 

1.  2  sq.  ft.  ?     3  sq.  ft.  ?     5  sq.  ft.  ?     10  sq.  ft.  ?     15 

sq.  ft.  ? 

2.  J  sq.  ft.  ?      J  sq.  ft.  ?      J  sq.  ft.  ?      £  sq.  ft.  ?     f 

sq.ft.? 


DENOMINATE  NUMBERS 


223 


f  sq.  yd.  ?     f  sq.  yd.  ? 


4  A.?     5  A.?     10  A.? 

A  A.?    f  A.?     JA.? 


2.  How  many  square  feet  in  : 

1.  5  sq.  yds.?     10  sq.  yds.?     25  sq.  yds.?     90  sq. 

yds.  ?     100  sq.  yds.  ? 

2.  144  sq.  in.  ?    288  sq.  in.  ?    432  sq.  in.  ?    720  sq. 

in.  ?     1440  sq.  in.  ? 

3.  J  sq.  yd.  ?     J  sq.  yd.  ? 

f  sq.  yd.  ? 

3.  How  many  square  rods  in  ? 

1.  1  A.?     2  A.?     3  A.? 

2.  J  A.  ?     J  A.  ?     1  A.  ? 

4.  How  many  square  yards  in  9  sq.  ft.  ?     45  sq.  ft.  ? 
108  sq.  ft.  ?     144  sq.  ft.  ?     1728  sq.  ft.  ? 

5.  How  many  acres  in  : 

1.  160  sq.  rds.  ?    320  sq.  rds.  ?    640  sq.  yds.  ?    1600 

sq.  yds.  ?     8000  sq.  yds.  ? 

2.  1  sq.  mi.  ?     ^  sq.  mi.  ?     1  sq.  mi.  ?     ^  sq.  mi.  ? 

T%  sq.  mi.  ? 

6.  How  many  square  feet  in  : 

1.  3  sq.  ft.  and  6  ft.  ?     5  sq.  yds.  and  3J  ft. 

2.  4  sq.  ft.  and  72  sq.  in.  ?     6  sq.  ft.  and  144  sq.  in.  ? 

VOLUME    MEASURE. 

1.  Volume  or  Cubic  Measure  is  used  in  measuring 
that  which  has  length,  breadth,  and  thickness. 

2.  The  volume  of  a  body  is  called  its  Solid  Contents, 
and  the  body  is  called  a  Solid. 

3.  A  solid  with  six  equal  square  faces  is 
called  a  Cube. 


"When  the  faces  are  square  inches,  the  solid  is  a  cubic 
inch. 

When  the  faces  are  square  feet,  the  solid  is  a  cubic 
•foot. 


Cube. 


224  ELEMENTARY  ARITHMETIC 

Table. 


1728  Cubic  Inches  (cu.  in.)  =  1  Cubic  Foot  (cu.  ft.). 
27  Cubic  Feet  =  1  Cubic  Yard  (cu.  yd.). 

128  Cubic  Feet  =  1  Cord  (cd.)  of  wood. 

24f  Cubic  Feet  =  1  Perch  of  stone. 


ORAL    EXERCISES. 
1.  How  many  cubic  inches  in  : 

1.  1  cu.  ft.  ?     2  cu.  ft.  ?     3  cu.  ft.  ?     5  cu.  ft.  ?     10 

cu.  ft.  ? 

2.  £  cu.  ft.  ?     \  cu.  ft.  ?     £  cu.  ft.  ?     ^  cu.  ft.  ? 

' 


2.  How  many  cubic  feet  in  : 

1.  led.?     2cd.?     3cd.?     5cd.?     10  cd.  ? 

2.  1  cu.  yd.  ?     3  cu.  yd.  ?     5  cu.  yd.  ?     7  cu.  yd.  ? 

10  cu.  yd.  ? 

3.  How  many  cu.  yards  in  27  cu.  ft.  ?     54  cu.  ft.  ?     81 
cu.  ft.  ?     108  cu.  ft.  ?     2700  cu.  ft.  ? 

4.  How  many  cu.  inches  in  1  cu.  ft.  and  144  cu.  in.  ? 

2  cu.  ft.  1000  cu.  in.  ? 

5.  How  many  cu.  feet  in  3  cu.  yd.  5  cu.  ft  ?     5  cu.  yd. 

3  cu.  ft.  ? 

LIQUID    MEASURE. 
Liquid  Measure  is  used  in  measuring  liquids. 

Table. 


4  Gills  (gi.)  =  1  Pint  (pt.). 
2  Pints          =  1  Quart  (qt.). 
4  Quarts        =  1  Gallon  (gal.). 


12  pt.? 
fpt.? 


DENOMINATE  NUMBERS  225 

31£  Gallons  =  1  Barrel  (bbl.). 
63    Gallons  =  1  Hogshead  (hhd.). 
1    Gallon   =  231  Cubic  Inches. 

ORAL    EXERCISES. 

1.  How  many  gills  in  : 

1.  1  pt,?     3pt?     5pt.?     8  pt? 

2.  |  pt.  ?     J  pt.  ?     }  pt.  ?     £  pt.  ? 

2.  How  many  pints  in  : 

1.  1  qt.  ?     3  qt.  ?     8  qt.  ?     10  qt.  ?     40  qt.  ? 

2.  3  qt.  1  pt.  ?     5  qt.  1  pt.  ?     9  qt.  2  pt.  ?     J  gal.  ? 

Jgal.? 

3.  How  many  quarts  in  : 

1.  1  gal.  ?     5  gal.  ?     6  gal.  ?     10  gal.  ?     25  gal.  ? 

2.  2  pt.  ?     8  pt.  ?     16  pt.  ?     20  pt.  ?     100  pt,  ? 

4.  How  many  gallons  in : 

.1.  4  qt.?  "  12  qt?  16  qt  ?  24  qt  ?  40  qt  ? 

2.  8  pt  ?  16  pt  ?  20  pt  ?  56  pt.  ?  100  pt.  ? 

3.  1  bbl.  ?  2  bbl.  ?  1  hhd.  ?  2  hhd.  ? 

5.  How  many  cubic  inches  in  a  gallon  ?     In  2  gal.  ? 


DRY    MEASURE. 

Dry  Measure  is  used  in  measuring  grain,  fruit,  and 
vegetables. 

Table. 


2  Pints  (pt.)  =  1  Quart  (qt.). 
8  Quarts  =  1  Peck  (pk.). 
4  Pecks  =  1  Bushel  (bu.). 


1  Bushel  =  2150.42  cubic  inches. 
15 


226  ELEMENTARY  ARITHMETIC 

ORAL    EXERCISES. 

1.  How  many  pints  in  4  qt.  ?     5  qt.  ?     7  qt.  ?     8  qt.  ? 
10  qt.  ? 

2.  How  many  quarts  in  : 

1.  2pk.?     4pk.?     9pk.?     10  pk.  ?     25  pk.? 

2.  2pt.?     12  pt.?     16  pt.?     20  pt.?     30  pt.  ? 

3.  How  many  pecks  in  : 

1.  1  bu.  ?     3  bu.  ?     5  bu.  ?     8  bu.  ?     40  bu.  ? 

2.  I' bu.  ?    \  bu.  ?    |  bu.  ?     £  bu.  ?    £  bu.  ? 

4.  How  many  busbels   in  4  pk.  ?     16  pk.  ?     20   pk.  ? 
24  pk.  ?     36  pk.  ? 

5.  How  many  pecks  in  8  bu.  3  pk.  ?     In  96  qt.  ? 

6.  How  many  cubic  inches  in  1  bu.  ?     In  2  bu.  ? 

AVOIRDUPOIS   WEIGHT. 

Avoirdupois  Weight  is  used  in  weighing  heavy  articles, 
except  gold  and  silver. 

Table. 


16  ounces  (oz.)  =  1  pound  (lb.). 

100  Pounds  =  1  hundred- weight  (cwt.). 

20  hundred-weight  )  =  j  T) 

2000  pounds  j 


2240  pounds  =  1  long  ton. 

ORAL    EXERCISES. 
1.  How  many  ounces  in  : 

1.  lib.?     21b.?     31b.?     41b.?     5Jlb.? 

2.  1  cwt.  ?     10  cwt.  ?    IT.?     IT.?    A  T.  ? 


DENOMINATE  NUMBERS 


227 


2.  How  many  pounds  in  : 

1.  1  cwt.  ?     3  cwt.  ?    4J  cwt.  ?    6  cwt.  ?    10  cwt.  ? 

2.  IT.?    3T.?    3JT.?     7T.?     10JT.? 

3.  I  T.  ?    IT.?    £  T.  ?    |  T.  ?    ^  T.  ? 

3.  How  many  hundred-weight  in  100  Ib.  ?     300  lb.? 
400  lb.?     450  Ib.  ?     lOOOlb.? 

4.  How  many  tons  in  1000  lb.  ?     2500  lb.  ?     3000  lb.  ? 
4250  lb.  ?     10,000  lb.  ? 

5.  How  many  long  tons  in  2240  lb.  ?    In  6720  lb.  ? 


TROY  WEIGHT. 
x  Table. 


24  Grains  (gr.)     =  1  Pennyweight  (pwt.). 
20  Pennyweight  =  1  ounce  (oz.). 
12  ounces  =  1  pound  (lh.). 


1  Troy  lb.  =  5760  gr. 

1  Avoirdupois  lb.  =  7000  gr. 
1  Troy  oz.  =  480  gr. 

1  Avoirdupois  oz.  •=  437^  gr. 


ORAL    EXERCISES. 

1.  How  many  grains  in: 

1.  1  pwt.  ?    2  pwt.  ?    5  pwt.  ?    15  pwt.  ?    20  pwt.  ? 

2.  1  oz.  ?     2  oz.  ?     3  oz.  ?     |  oz.  ?     f  oz.  ? 

2.  How  many  pennyweight  in  : 

1.  1  oz.  ?     2  oz.  ?     6  oz.  ?     7  oz.  ?     8J  oz.  ? 

2.  24  gr.  ?     48  gr.  ?     72  gr.  ?     96  gr.  ?     100  gr.  ? 


228  ELEMENTAKY  ARITHMETIC 

3.  How  many  ounces  in : 

1.  lib.?    41b.?    4|lb.?     51b.?    6|  lb.? 

2.  20pwt?   40pwt?   50pwt.?    60pwt.?   TOpwt.? 

3.  3lb.  3oz.?    41b.4oz.?    5  Ib.  5  oz.  ?    6lb.6oz.? 

4.  How  many  pounds  in  12  oz.  ?     24  oz.  ?     36  oz.  ? 
84  oz.  ?     144  oz.  ? 

5.  How  many  gr.  in  1  troy  Ib.  ?     1  avoirdupois  Ib.  ? 

6.  How  many  gr.  in  1  troy  oz.  ?     1  avoirdupois  oz.  ? 

APOTHECARIES'   WEIGHT. 

Apothecaries'  Weight  is  used  in  weighing  medicines 
required  by  prescriptions. 

Table. 


20  Grains  (gr.)  =  1  Scruple  (sc.  or  ^) 
3  Scruples        =  1  Dram  (dr.  or  3). 
8  Drams  =  I  ounce  (oz.  or  ^). 

12  Ounces  =  1  pound  (Ib.  or  R>). 


ORAL    EXERCISES. 

1.  How  many  grains  in  1  sc.  ?     4  sc.  ?    5  sc.  ?    5^  sc.  ? 

2.  How  many  scruples  in  : 

1.1  dr.?     3dr.?     6  dr.  ?     10  dr.  ?     lOJ-dr.? 
2.  20  gr.?     40  gr.?     60  gr. ?     70  gr.?     HOgr.? 

3.  How  many  drams  in  : 

1.  1  oz.  ?     4  oz.  ?     6  oz.  ?     8£  oz.  ?     9£  oz.  ? 

2.  39?     99?     249?     369?     409? 

4.  How  many  ounces  in  1  Ib.  ?    3  Ib.  ?    5|  Ib.  ?    6£  Ib.  ? 
10}  Ib.  ? 

5.  How  many  pounds  in  12  oz.  ?    60  oz.  ?    96  oz.  ?    100 
oz.?     150oz.? 


DENOMINATE  NUMBERS 


229 


DIVISIONS  OF  TIME. 
Table. 


60  Seconds  (sec.)  =  1  Minute  (min.). 


60  Minutes 
24  Hours 
7  Days 

365  Days 

366  Days 
100  Years 


=  1  Hour  (hr.). 

=  1  Day  (da.). 

=  1  Week  (wk.). 

=  1  Year  (yr.). 

==  1  Leap  Year  (1.  yr.). 

=  1  Century  (C.). 


Centennial  years  exactly  divisible  by  400,   and   other  years  exactly 
divisible  by  4,  are  leap  years. 

12  Months  =  1  Year  (yr.). 

Table. 


1.  January  (Jan.)  =31  da. 

2.  February  (Feb.)  —  28  or  29*  da. 

3.  March  (Mar.)      =  31  da. 

4.  April  (Apr.)        =  30  da. 

5.  May  (May)         =  31  da. 

6.  June  (June)        =  30  da. 


7.  July  (July)  =  31  da. 

8.  August  (Aug  )       =  31  da. 

9.  September  (Sept.)  =  30  da, 

10.  October  (Oct.)        =  31  da. 

11.  November  (Nov.)  =  30  da. 

12.  December  (Dec.)   =  31  da. 


A  Useful  Rhyme. 
Thirty  days  hath  September, 
April,  June,  and  November. 
All  the  rest  have  thirty-one, 
Excepting  February,  which  stands  alone 
With  twenty-eight,  till  one  day  more 
We  add  to  it  one  year  in  four. 


One  day  added  to  make  leap  year. 


230 


ELEMENTARY  ARITHMETIC 


ORAL    EXERCISES. 

1.  How  many  seconds  in  1  rain.?     3  min. ?    8  rain.? 
12  min.  ?     50  min.  ? 

2.  How  many  minutes  in  : 

1.  1  hr.  ?     2  hr.  ?     9  hr.  ?     13  hr.  ?     15  hr.  ? 

2.  60  sec.  ?     120  sec.  ?     200  sec.  ?    300  sec.  ?     500 

sec.  ? 

3.  How  many  hours  in  : 

1.  1  da.  ?     4  da,  ?     10  da.  ?     20  da.  ?     30  da.  ? 

2.  -I- da.?    fda.?    fda.?    |Jda.?    £da.? 

3.  60  min.  ?     240  min.  ?     300  min.  ?     1200  min.  ? 

1800  min.  ? 

4.  How  many  days  in  : 

1.  1  wk.  ?     7  wk.  ?     9  wk,  ?     12  wk.  ?     52  wk.  ? 

2.  3  wk.  3  da.?     4  wk.  2  da.?    4  wk.  3  da.? 

iwk.? 

5.  Which  of  these  are  leap  years:  1492?    1500?    1600? 
1876?     1892?     1898?     1900?     2000? 

6.  Name  the  four  months  that  have  30  da.  each. 

7.  Name  the  month  that  changes  the  number  of  its 
days.     Give  a  reason  for  the  change. 

COUNTING. 
The  following  denominations  are  frequently  used  in 

counting : 

Table. 


12  things  =  1  dozen  (doz.). 
12  dozen  =  1  gross  (gr.). 
12  gross    =  1  great  gross  (G.  gr.). 
20  things  =  1  score. 


DENOMINATE  NUMBERS 
Stationers'  Table. 


231 


24  sheets     =  1  quire  (qr.). 
20  quires     =  1  ream  (R.) 

2  reams     =  1  bundle. 

5  bundles  =  1  bale. 


ORAL  EXERCISES. 

1.  How  many  single  things  in  : 

1.  Idoz.?    5doz.?    12doz.?    15  doz..?    30doz.? 

2.  J  doz.  ?     J  doz.  ?     f  doz.  ?     £  doz.  ?     £  doz.  ? 

3.  1  score?     2  score?     8  score?     10  score?     12 

score  ? 

2.  How  many  things  in  : 

1.  Igr.?     5gr.?     9gr.?     12  gr.  ?     20  gr.? 

2.  f  score?    f  score?    f  score?     ^  score?     y 

score  ? 

3.  1  G.  gr.?     3  G.  gr.?     5  G.  gr.?     £  G.  gr.  ? 


. 

3.  How  many  sheets  in  1  quire  ?    2 
R.  ?     1  bundle  ?     5  bundles  ?     1  bale  ? 


20  quires  ?    1  R.  ?    2 


REVIEW. 
Find  the  value  of  x  in  the  following  equations  : 

The  sign  .-.  means  therefore. 


1.  3  ft.  =  x  in. 

1  ft.  =  12  in. 

.-.  x  =  S  X  12  in.  =36. 

2.  96  in.  =  x  ft. 

12  in.  =  1  ft. 

.-.  a?  ft.  =*ftft.  =  8  ft. 


3.  64  qt.  =  x  pk. 

4.  f  A.  —  x  sq.  rd. 

5.  5  yd.  =  x  in. 

6.  96  qt.  ==  x  gal. 

7.  -y1  hr.  =  x  min. 

8.  5  R.  —  x  qr. 


232  ELEMENTARY  ARITHMETIC 

9.  6  Ib.  —  x  oz.  18.  f  cu.  ft.  =  x  cu.  in. 

10.  5  bu.  =  x  pk.  19.  144  gr.  —  x  pwt. 

11.  12  qt.  =  x  pt.  20.  72  oz.  =  x  Ib. 

12.  7  oz.  =  x  pwt.  21.  5  score  =  x  units. 

13.  27  sq.  ft.  =  x  sq.  yd.  22.  2  gross  =  x  doz. 

14.  140  da.  =  x  wk.  23.  J  G.  gr.  =  x  doz. 

15.  120  items  =  x  doz.  24.  $8  =  x  cents. 

16.  6  qr.  —  x  sheets.  25.  $f  —  x  cents. 

17.  81  cu.  ft.  =  x  cu.  yd.  26.  1000  mills  =  jx. 

REDUCTION   OF  DENOMINATE  NUMBERS. 

DEFINITIONS. 

1.  Reduction  changes  a  denominate  numher  from  one 
denomination  to  another  without  altering  its  value. 

2.  Reduction  Descending  changes  a  number  from  a 
higher  to  a  lower  denomination. 

3.  Reduction  Ascending  changes   a  number   from  a 
lower  to  a  higher  denomination. 

Reduction  Descending-. 

EXERCISES. 
1.  How  many  inches  are  there  in  8  yd.  2  ft.  5  in.  ? 

Process. 
8  yd.  2  ft.  5  in.  Explanation. 

X  3  1  yd.  =  8  ft. 

~~24  ,       •*•  8  yd.  =  8  X  3  ft.  =  24  ft. 

24  ft.  +  2  ft.  =  26  ft. 
1  ft.  =  12  in. 

.-.  26  ft.  =  26  X  12  in.  =  312  in. 

X  12  312  in.  +  5  in.  ==  317  in. 

312  Therefore,  8  yd.  2  ft.  6  in.  =  317  in. 

+  5 
317 


REDUCTION  OF  DENOMINATE  NUMBERS  233 

2.  Reduce  to  lower  denominations  the  following : 

1.  5  yd.  3  ft.  9  in.  11.  5  hr.  26  min.  26  sec. 

2.  13  yd.  2  ft.  8  in.  12.  21  hr.  23  min.  29  sec. 

3.  16  yd.  2  ft.  11  in.  13.  29  hr.  30  min.  46  sec. 

4.  38  yd.  2  ft.  10  in.  14.  40  hr.  40  min.  40  sec. 

5.  49  yd.  1  ft.  7  in.  15.  5  cwt.  41  Ib.  9  oz. 

6.  7  gal.  3  qt.  1  pt.  16.  9  cwt.  86  Ib.  13  oz. 

7.  14  gal.  2  qt.  1  pt.  17.  3  T.  6  cwt.  90  Ib.  13  oz. 

8.  26  gal.  3  qt.  1  pt.  18.  5  T.  8  cwt.  46  Ib.  10  oz. 

9.  29  gal.  3  qt.  1  pt.  19.  10  T.  10  cwt.  10  Ib.  10  oz. 
10.  58  gal.  1  qt.  1  pt,  20.  9  Ib.  5  oz.  11  pwt.  5  gr. 

3.  Reduce  to  lower  denominations  the  following : 

1.  6  Ibs.  6  oz.  17  pwt.  10  gr. 

2.  11  Ib.  11  oz.  11  pwt.  11  gr. 

3.  4  bu.  4  pk.  6  qt.  1  pt. 

4.  6  bu.  3  pk.  7  qt.  J  pt. 

5.  7  bu.  2  pk.  5  qt.  1  pt. 

6.  100  bu.  3  pk.  7  qt.  1  pt. 

7.  5  rd.  4  yd.  2  ft.  7  in. 

8.  6  rd.  3  yd.  2  ft.  2  in. 

9.  9  rd.  2yd.  2  ft.  11  in. 

10.  18  rd.  1  yd.  1  ft.  1  in. 

11.  4  sq.  yd.  5  sq.  ft.  19  sq.  in. 

12.  6  sq.  yd.  7  sq.  ft,  99  sq.  in. 

13.  10  sq.  yd.  6  sq.  ft.  141  sq.  in. 

14.  3  A.  120'sq.  rd.  6  sq.  yd. 

15.  9  A.  36  sq.  rd.  25  sq.  yd. 

16.  18  A.  72  sq.  rd.  30  sq.  yd. 

17.  3  cu.  yd.  11  cu.  ft.  96  cu.  in. 

18.  5  cu.  yd.  7  cu.  ft.  825  cu.  in. 

19.  10  cu.  yd.  10  cu.  ft.  10  cu.  in. 


234  ELEMENTARY  ARITHMETIC 

20.  4  lb.  4  oz.  4  dr.  29. 

21.  51b.  9  oz.  6  dr.  19. 

22.  10  lb.  10  oz.  7  dr.  2  sc. 

23.  4  R.  11  qr.  19  sheets. 

24.  5  R  9  qr.  21  sheets. 

25.  4  lb.  7  oz.  11  pwt. 

26.  6  lb.  11  oz.  4  sc. 

27.  7  da.  11  hr.  36  min. 

28.  9  rd.  5  yd.  2  ft.  11  in. 

29.  6  A.  151  sq.  rd.  29  sq.  yd. 

30.  8  mi.  211  rd.  16  ft.  6  in. 

4.  Reduce  .85  yd.  to  feet  and  inches. 
Process.  Explanation. 

.85  yd.  1  yd.  =  3  ft. 

3  .85  yd.  =  .85  of  3  ft.  =  2.55  ft. 

2^5  ft.  1  ft-  =   12  in. 

12  .65  ft.  =-.56  of  12  in.  =6.6  in. 


6.60  in.  .  '.  85  yd.  =  2  ft.  6.6  in. 

5.  f  cwt.  equal  how  many  pounds  and  ounces  ? 

Process.  Explanation. 

|  X  100  ==  *jp-  =  62£.  l  cwt.  =  100  lb. 

1Xl6_JL6-  —  8  f  cwt.  —  |  of  100  lb.  =  62 

!-.|cwt.=2621b.'8oz.  "i^16;^ 

^  lb  =  |  of  16  oz.  =  8  oz. 
.-.  £  cwt.  =  62  lb.  8  oz. 

Or, 

5  ___    (J25  f  cwt.  —  .625  cwt. 

8 

1  CWt.  =  100  lb. 


g2.500  -625  cwt-  =  -625  of  10°  lb-  =  62-5 

IQ  I  lb.  =  16  oz. 

-6  lb-  =  -5  of  16  oz.  =  8  oz. 


/.  |  cwt.  =  62  lb.  8  oz. 


REDUCTION  OF  DENOMINATE  NUMBERS 


235 


6.  Reduce  to  integers  of  lower  denominations : 


1.  |  rd. 

2.  |  wk. 

3.  f  yd. 

4.  .7  bu. 

5.  .56  T. 

6.  .5  cu.  yd. 

7.  %  mi. 

8.  f  A. 

9.  .84  wk. 

10.  .4236  gal. 

11.  .585  yd. 

12.  .625  bundle. 

13.  I  G.  gr. 

14.  |  Ib.  (Apoth.) 

15.  ^lb.(Troy.) 


16.  |  T. 

17.  |  bu. 

18.  |  gal. 

19.  .3456  cd. 

20.  .1234  A. 

21.  \\  mi. 

22.  f  sq.  mi. 

23.  T\  cu.  yd. 

24.  .875  gal. 

25.  .4bu. 

26.  .Of  oz.  (Troy.) 

27.  .001  Ib.  (Apoth.) 

28.  .0007  C.  (Time.) 

29.  .9009  score. 

30.  .MA- bale. 


Reduction   Ascending-. 

EXERCISES. 
1.  Eeduce  345  pints  to  bushels,  pecks  and  quarts. 


Process. 


2 
8 
4 

345  pt. 
172  qt.  1  pt. 
21  pk.  4  qt. 

5  bu.  1  pk. 

.-.  345  pt.  =  5  bu. 
1  pk.  4  qt.  1  pt. 


Explanation. 

2  pt.  •-=  1  qt.  .'.  £  the  number  of  pints 
=  the  number  of  quarts. 

345  pt.  =  ifa.  qt.  =  172  qt.  1  pt. 

8  qt.  =  1  pk.  .-.  |  the  number  of  quarts 
=  the  number  of  pk. 

172  qt.  =  ip  pk.  =  21  pk.  4  qt. 

4  pk.  —  1  bu.  .-.  \  the  number  of  pecks 
=  the  number  of  bu. 

21  pk.  =  *£  bu.  =  5  bu.  1  pk. 


Hence,  345  pt.  =  5  bu.  1  pk.  4  qt.  1  pt. 


236  ELEMENTARY  ARITHMETIC 

2.  Reduce  4392  inches  to  rods,  yards,  etc. 


Process. 


12 


2 
11 


4392  in. 


366  ft.  0  in. 


122  yd.  0  ft. 
2 


244  half  yd. 
22  rd.  2  half  yd. 
or  1  yd. 

•.  4392  in.  =  22  rd.  1  yd. 


Hence,  4392  in.  =  22  rd.  1  yd. 


3.  Reduce: 

1.  5324 

2.  4296 

3.  6835 

4.  4640 

5.  3567 

6.  2706 

7.  4675 

8.  5794 

4.  Reduce: 

1.  4054 

2.  6048 

3.  2905 

4.  3426 

5.  8975 

6.  7865 


gi.  to  gal.,  etc. 
pt.  to  gal.,  etc. 
gi.  to  gal.,  etc. 
in.  to  rd.,  etc. 
in.  to  rd.,  etc. 
in.  to  rd.,  etc. 
pt.  to  bu.,  etc. 
pt.  to  bu.,  etc. 


Explanation. 

12  in  =  1  ft.  .-.  jf  the  number 
of  in.  =  the  number  of  feet. 

4392  in.  =  366  ft. 

3  ft.  =  1  yd.  .  •.  £  the  number 
of  feet  =  the  number  of  yd. 

366  ft.  =  *f  *  yd.  =  122  yd. 

6£  yd.  =11  half  yd.  =  1  rd. 
.  •.  ^  the  number  of  half  yd.  = 
the  number  of  rods. 

£44.  yd.  =  jft  ra.  =  22  rd.  2 
half  yd. 

2  half  yd.  =  1  yd. 


9.  4058  pt.  to  bu.,  etc. 

10.  6275  oz.  to  cwt.,  etc. 

11.  9238  oz.  to  cwt,  etc. 

12.  6094  Ib.  to  tons,  etc. 

13.  4806  Ib.  to  tons,  etc. 

14.  5396  hr.  to  wk.,  etc. 

15.  9279  sec.  to  hr.,  etc. 

16.  10,905  min.  to  da.,  etc. 


sq.  in.  to  sq.  yd.,  etc. 
sq.  in.  to  sq.  yd.,  etc. 
sheets  to  R.,  etc. 
sheets  to  R.,  etc. 
cu.  in.  to  cu.  ft.,  etc. 
gr.  to  Ib.  (troy),  etc. 


REDUCTION  OF  DENOMINATE  NUMBERS  237 

7.  9497  gr.  to  Ib.  (troy),  etc. 

8.  10,249  gr.  to  Ib.  (apoth.),  etc. 

9.  876,000  hr.  to  C. 
10.  392,040  sq.  ft.  to  A. 

5.  Reduce  6  oz.  6  pwt.  to  the  fraction  of  1  Ib. 

Process.  Explanation. 

6  OZ.  6  pwt.      1  Ib.  1  lb.  =  12  x  20  ==  240  pwt. 

20  12  6  oz.  6  pwt.  =  126  pwt. 

126  pwt.  12  oz.  l  pwt.  =  ^  of  1  lb. 

20  •••  126  pwt.  =  $f$  of  llb.  =  f£of  lib. 


=  JJ-.  240  pwt.  Hence,  6  oz.  6  pwt.  =  f  £  of  a  lb. 


6.  Reduce: 

1.  2  ft.  4  in.  to  the  fraction  of  a  yard. 

2.  4  qt.  1  pt.  to  the  fraction  of  a  peck. 

3.  5  pk.  4  qt.  to  the  fraction  of  a  bushel. 

4.  3  qt.  1  pt.  3  gi.  to  the  fraction  of  a  gallon. 

5.  3  pk.  5  qt.  1  pt.  to  the  fraction  of  a  bushel. 

6.  5  yd.  2  ft.  7  in.  to  the  fraction  of  a  rod. 

7.  5  da.  6  hr.  to  the  fraction  of  a  week. 

8.  15  hr.  12   min.    18   sec.  to   the    fraction  of  a 

day. 

9.  2  cwt.  5  lb.  12  oz.  to  the  fraction  of  a  ton. 

10.  5  qt.  1  pt.  to  the  fraction  of  a  bushel. 

11.  3  gal.  4  qt.  1  pt.  3  gi.  to  the  fraction  of  a  barrel. 

12.  $52  cents  5  mills  to  the  fraction  of  an  eagle. 

13.  6  oz.  6  pwt.  6  gr.  to  the  fraction  of  a  lb.     Of  2 

lb.     Of  5  lb. 

14.  360  da.  3  wk.  3  da.  4  hr.  to  the  fraction  of  a 

year.     Of  10  years. 


238  ELEMENTARY  ARITHMETIC 

ADDITION  OF  DENOMINATE  NUMBERS. 

EXERCISES. 

1.  Find  the  sum  of  4  gal.  4  qt.  1  pt. ;  6  gal.  3  qt.  1  pt. ; 
7  gal.  3  qt.  1  pt. ;  9  gal.  2  qt. 

Process.  Explanation. 

Gal.         Qt.        Pt.  1.  PRINCIPLE  :  Only  units  of  like  order  can  be 

441  added. 

c        •  o  i  2.  Numbers  of  like  denomination  are  written 

o          o          I 

in  the  same  column. 

3.  The  sum  of  the  pints  is  3  pt.  =  1  qt.  1  pt. 
9  2 0_  4.  We  write  the  1  pt.  and  reserve  the  1  qt. 


2911  5.  The  sum  of  the  1  qt.  and  the  column  of  qt. 

is  13  qt.  =  3  gal.  1  qt. 

6.  The  sum  of  the  3  gal.  and  the  column  of  gal.  is  29  gal. 
Hence,  the  sum  total  is  29  gal.  1  qt.  1  pt. 

2.  Find  the  sum  of: 

1.  4  gal.  3  qt.  1  pt. ;  28  gal.  2  qt.  1  pt. ;  29  gal.  3  qt. 

2.  9  bu.  2  pk.  6  qt. ;  27  bu.  3  pk.  6  qt. ;  23  bu.  2  pk. 

5  qt. 

3.  8  da.  6  hr.  31  min. ;  9  da.  25  hr.  21  min. ;  7  da. 

29  hr. 

4.  35  Ib.  7  oz.  (avoirdupois);  46  Ib.  15  oz. ;  37  Ib. 

13  oz.;  94  Ib. 

5.  24  Ib.  5  oz.  19  pwt.  9  gr.;  22  Ib.  6  oz.  18  pwt. 

21  gr. ;  21  Ib.  21  gr. 

6.  21  yd.  2  ft.  11  in. ;  26  yd.  2  ft.  10  in. ;  9  yd.  1  ft. 

8  in. ;  29  yd.  2  ft.  9  in. 

7.  26  gal.  2  qt.  1  pt.  2  gi. ;  29  gal.  3  qt.  1  pt.  1  gi. ; 

39  gal.  3  qt. ;  32  gal. 

8.  23  A.  46  sq.  rd.;  25  A.  120  sq.  rd.;  26  A.  143 

sq.  rd. ;  22  A.  107  sq.  rd. 


SUBTEACTION  OF  DENOMINATE  NUMBEKS          239 

9.  6  T.  7  cwt.  25  Ib.  11  oz. ;  8  T.  6  cwt.  47  Ib.  14 

oz. ;  28  T.  95  Ib. 

10.  6  yd.  1  ft.  9  in. ;  5  yd.  2  ft.  9  in. ;  5  yd.  1  ft.  11 
in. ;  4  yd.  2  ft. ;  4  yd. 

SUBTRACTION  OF  DENOMINATE  NUMBERS. 

1.  From  26  gal.  2  qt.  1  pt.  2  gi.  take  19  gal.  3  qt.  0  pt. 

3gi. 

Process.  Explanation. 

Gal.         Qt.        Pt.        Gi.  1.  PRINCIPLE  :   Only  like  num- 

2(3           2           1           2  bers  can  be  subtracted. 

iq           o           n           Q  2.  We   write    the    numbers   of 

—  the  same  denomination  in  the  same 


6  column 

3.'  We  hegin  to  subtract  at  the  lowest  denomination. 

4.  3  gi.  cannot  be  taken  from"  2  gi. 

5.  We  therefore  add  the  1  pt.,  or  4  gi.,  to  the  2^gi.,  making  6  gi.,  and 
then  say  6  gi.  —  3  gi.  =  3  gi. 

6.  The  1  pt.  having  been  used,  0  pt.  —  0  pt.  =  0  pt. 

7.  3  qt.  cannot  be  taken  from  2  qt. 

8.  Adding  1  gal.,  or  4  qt.,  we  have  6  qt. 

9.  6  qt.  —  3  qt.  =  3  qt. 

10.  Finally,  25  gal.  —  19  gal.  =  6  gal. 

11.  Kemainder  =  6  gal.  3  qt.  0  pt.  3  gi. 

2.  From  9  bu.  3  pk.  5  qt.  take  4  bu.  3  pk.  6  qt. 

3.  From  9  bu.  0  pk.  4  qt.  take  4  bu.  3  pk.  6  qt. 

4.  From  13  gal.  3  qt.  1  pt.  3  gi.  take  6  gal.  4  qt.  0  pt. 

5.  From  11  da.  6  hr.  31  min.  take  8  da.  8  hr.  11  min. 

6.  From  42  Ib.  5  oz.  16  pwt.  take  9  Ib.  7  oz.  13  pwt. 

7.  From  26  yd.  1  ft.  8  in.  take  5  yd.  2  ft.  11  in. 

8.  'From  23  rd.  5  yd.  2  ft.  take  9  rd.  5  yd.  2  ft. 

9.  From  24  Ib.  9  oz.  6  dr.  take  8  Ib.  11  oz.  6  dr. 

•     10.  From  81  sq.  rd.  take  26  sq.  rd.  7  sq.  ft.  Ill  sq.  in. 


240  ELEMENTARY  ARITHMETIC 

11.  From  36  T.  9  cwt.  86  Ib.  11  oz.  take  21  T.  13  cwt. 
46  Ib.  13  oz. 

12.  From  132  gal.  1  qt.  1  pt.  1  gi.  take  128  gal.  3  qt. 
1  pt.  3  gi. 

DIFFERENCE    OF   DATES. 

1.  How  many  years,  months,  and  days  from  June  27, 
1853,  to  Dec.  25,  1898? 

Process.  Explanation. 

Yr.        Mo.     Da.  1.  Dec.  is  the  12th  mo.,  June  the  6th  mo. 

1898      12      25  2-  We  add  *  mo-  or  30  da-  to  25  da->  making 

1853       6     27          65da^s- 

—  TI  --  z  -  ^rr-  3.  We  then  subtract,  proceeding  as  with  other 

4O         O      Zo  , 

denominate  numbers. 

2.  How  long  was  it  between  : 

1.  Jan.  21,  1846,  and  June  26,  1898? 

2.  July  24,  1875,  and  Feb.  2,  1891  ? 

3.  Mar.  9,  1840,  and  Aug.  8,  1892? 

4.  Oct.  24,  1882,  and  Sept.  11,  1893? 

5.  Jan.  6,  1706,  and  Apr.  17,  1790? 


MULTIPLICATION  OF  DENOMINATE  NUMBERS. 

1.  Multiply  7  yd.  2  ft.  9  in.  by  6. 

Process.  Explanation. 

Yd.    Ft.    In.  1.  6  times  9  in.  =  54  in.  =  4  ft.  6  in. 

729  2.  "We  write  the  6  in.  and  reserve  the  4  ft. 

g  3.  6  times  2  ft.  =  12  ft.  ;  12  ft.  -f  4  ft.  reserved  = 

•^  -  j  -  g-  16  ft.  ==  5  yd.  1  ft. 

4.  We  write  the  1  ft.  and  reserve  the  5  yd. 
5.  6  times  7  yd.  =  42  yd.  ;  42  yd.  -f  5  yd.  =  47  yd. 
Hence,  the  product  is  47  yd.  1  ft.  6  in. 


DIVISION  OF  DENOMINATE  NUMBERS  241 

2.  Multiply: 

1.  5  bu.  2  pk.  6  qt.  by  8. 

2.  6  gal.  2  qt.  1  pt.  3  gi.  by  7. 

3.  6  Ib.  7  oz.  11  pwt.  9  gr.  by  9. 

4.  8  Ib.  9  oz.  6  dr.  29  11  gr.  by  8. 

5.  4  hr.  30  min.  46  sec.  by  6. 

6.  3  T.  6  cwt.  59  Ib.  14  oz.  by  9. 

7.  3  rd.  4  yd.  2  ft.  11  in.  by  10. 

8.  23  cu.  yd.  16  cu.  ft.  1226  cu.  in.  by  7. 

9.  9  sq.  yd.  3  sq.  ft.  56  sq.  in.  by  6. 

10.  6  da.  9  hr.  26  min.  36  sec.  by  5. 

11.  23  rd.  5  yd.  2  ft.  9  in.  by  9. 

12.  6  R.  9  qr.  17  sheets  by  10. 

13.  7  bbl.  11  gal.  2  qt.  1  pt.  by  8. 

14.  6  yr.  5  mo.  15  da.  18  hr.  by  5. 

15.  6  T.  12  cwi  95  Ib.  12  oz.  9  dr.  by  8. 

DIVISION   OF  DENOMINATE  NUMBERS. 

1.  Divide  45  bu.  3  pk.  1  qt.  by  8. 

Process.  Explanation, 

bu.    pk.    qt.  1.  |  of  45  bu.  =  5  bu.  and  5  bu.  remaining. 

8  )  45      3      1  2.  5  bu.  rem.  =  20  pk. ;  20  pk.  +  3  pk.  =  23  pk, 

r- 2      71  3-  i  °f  23  pk.  =  2  pk.,  and  7  pk.  remaining. 

4.  7  pk.  =  56  qt. ;  56  qt.  -f  1  qt.  =  67  qt. 
5.  $•  of  57  qt.  ==  7£  qt. 
Hence,  the  quotient  is  5  bu.  2  pk.  7|  qt. 

2.  Divide: 

1.  32  gal.  2  qt.  1  pt.  2  gi.  by  7.  , 

2.  26  bu.  3  pk.  6  qt.  1  pt.  by  5. 

3.  24  yd.  2  ft.  7  in.  by  8. 

4.  34  cwt.  79  Ib.  11  oz.  by  6. 

16 


242 


ELEMENTARY  ARITHMETIC 


5.  53  lb.  9  oz.  16  pwt.  by  10. 

6.  33  lb.  8  oz.  6  dr.  29  by  9. 

7.  20  br.  11  min.  47  sec.  by  7. 

8.  32  sq.  yd.  8  sq.  ft.  56  sq.  in.  by  5. 

9.  26  rd.  5  yd.  2  ft.  9  in.  by  6. 

10.  16  bbl.  11  gal.  2  qt.  1  pt.  by  8. 

11.  144  bu.  2  pk.  6  qt.  by  7. 

12.  42  gal.  1  qt.  1  pt.  1  gi.  by  12. 

13.  17  mi.  100  rd.  13  ft.  6  in.  by  11. 

14.  15  A.  140  sq.  rd.  3  sq.  yd.  by  3. 

15.  12  T.  5  cwt.  80  lb.  12  oz.  by  7. 

MEASUREMENT    OP    SURFACES. 

1.  A  square  has  been  defined. 

2.  Describe  a  square  inch.     Describe  a  square  foot. 

3.  A  figure  2  inches  long  and  1  inch  wide,  with  all  its 
angles  equal,  contains  how  many  square  inches  ? 

3  4.  A  figure  2  inches  long  and  2 

inches    wide    contains    how    many 
square  inches  ? 

5.  That  is  2,  the  length,  x  2,  the 
width,  =  what  ? 

Rectangle.  Q    A  figure   3  in<  ]ong  and  2  in. 

wide  contains  how  many  square  inches  ? 

7.  That  is  3,  the  length,  X  2,  the  width,  =  what? 

8.  The  number  of  square  units  in  a  figure  equals  the 
product  of  what  ? 

9.  The  number  of  square  units  in  a  figure  is  called  its 

Area. 

10.  A  figure  with  four  sides  and  four  right  angles  is 

called  a  Rectangle. 


MEASUREMENT  OF  SUEFACES  243 


PRINCIPLE. 

The  area  of  a  rectangular  surface  is  the  product 
of  its  length  and  width. 


FORMULA. 
Area  of  a  Rectangle  =  Length  X  Width. 

PROBLEMS. 

1.  Find  the  area  of  a  rectangular  lot  whose  length  is 
33  ft.  and  width  6  ft.  6  in. 

Process.  Explanation. 

33  x  gl  =  214^.  Area  —  Length  X  Width. 

•.  Area  =2144  sq.  ft.  The  P™1"01  of  38  and 

Hence,  the  area  is  214£  sq.  ft. 

The  process  requires  that  the  length  and  width  be  expressed  in  the  same 
denomination. 

2.  A  room  is  24  ft.  long  and  18  ft.  wide.     What  is  its 
area? 

3.  A  floor  is  10  ft.  wide  and  30  ft.  long.    Find  its  area. 

4.  The  side  wall  of  a  room  is  17  ft.  long  and  12  ft. 
high.     How  many  square  feet  in  the  wall  ?     How  many 
square  yards  ? 

5.  An  end  wall  of  a  room  is  12  ft.  by  12  ft.     How 
many  square  feet  in  the  wall  ?     How  many  square  yards  ? 

6.  I  have  a  rectangular  room.     Its  length  is  20  ft.,  its 
width,  18  ft.;  its  height,  12  ft.     Find  the  number  of 
square  feet  and  square  yards  in  the  four  walls. 

Suggestion  :  The  distance  around  the  room  X  the  height  =  the  area. 


244  ELEMENTARY  ARITHMETIC 

7.  A   rectangular   field   is,  in   length,  150   rods;   in 
breadth,  130  rods.     Find  the  number  of  square  rods  in 
the  field  and  also  the  number  of  acres. 

8.  A  room  is  25  ft.  long  and  15  ft.  wide.     Find  the 
area  of  both  ceiling  and  floor ;  also  the  cost  of  plastering 
the  ceiling,  at  25  cents  per  square  yard. 

9.  Find  the  cost  of  painting  the  walls  and  ceiling  of 
a  room  16  ft.  6  in.  long,  15  ft.  9  in.  wide,  14  ft.  high,  at 
25  cents  per  square  yard. 

10.  Find  the  area  of  a  square  field  whose  side  is  76 
rods. 

11.  Find  the  value  of  a  field  180  rds.  long  and  90  rds. 
wide,  at  $20  an  acre. 

12.  A  room  18  ft.  wide  and  24  ft.  long  was  covered 
with  carpet,  1  yd.  wide,  at  $1.00  per  yard.     How  much 
did  the  carpeting  cost  ? 

13.  I  have  a  rectangular  field  180.5  rds.  long  and  97.75 
rds.  wide.     How  many  acres  does  it  contain  ? 

14.  What  will  be  the  cost  of  cementing  the  floor  of  a 
cellar  56  ft.  by  45  ft.,  at  $.30  per  square  yard? 

15.  What  will  it  cost  to  pave  a  street  3  mi.  115  rds. 
long  and  2  rds.  wide,  at  $46.50  per  square  rod  ? 

16.  How  many  square  rods  in  a  garden  that  is  7  rods 
square  ? 

17.  How  many  acres  in  a  rectangular  field  36  rds.  12 
ft.  wide,  and  48  rds.  8  ft.  long  ? 

18.  What  is  the  cost  of  asphalting  a  walk  93|  ft.  by  6£ 
ft.,  at  $.75  per  square  yard? 

19.  A  side  of  a  public  square  is  660  ft.     How  many 
acres  does  it  contain  ? 

20.  If  a  carpet  is  36  in.  wide,  how  many  yards  of  it 


MEASUREMENT  OF  SOLIDS 


245 


will  be  required  to  carpet  a  room  18  ft.  by  16  ft?    What 
will  be  the  cost  at  $1.1 2|  per  yard  ? 

21.  Draw  a  figure  to  show  the  difference  between  6 
square  feet  and  6  feet  square. 


MEASUREMENT   OP    SOLIDS. 


A  cubic    foot.      A   cubic 


A  cube  has  been  defined. 

Describe    a    cubic    inch, 
yard. 

How  many  squares  in  the  base 
of  the  figure  ? 

We  will  call  them  square  feet. 

If  you  place  a  cubic  foot  of  wood 
or  stone  upon  each  square,  how 
many  cubic  feet  will  you  have  ? 

If  upon  those  you  place  12  more 
cubic  feet,  you  will  have  how  many 
cubic  feet.? 

If  you  add  a  third  layer  of  12  cubic  feet,  how  many 
will  you  have  ?  If  you  add  a  fourth  layer,  how  many  ? 
If  you  add  a  fifth  layer,  how  many  ? 

How  many  feet  high  is  your  structure  now  ? 

The  base  layer  has  how  many  cubes  ? 

Therefore,  the  number  12,  the  base,  X  5,  the  height, 
gives  you  what  ? 

A  Rectangular  Solid  stands  on  a  rectangular  base  and 
its  angles  are  all  right  angles. 

The  Volume  of  a  solid  is  the  number  of  cubic  units 
it  contains,  i.e.,  of  cubic  inches,  cubic  feet,  cubic  yards. 


246  ELEMENTARY  ARITHMETIC 


PRINCIPLE. 

The  volume  of  a  rectangular  solid  is  the  product 
of  its  length,  width,  and  height. 


FORMULA. 

Volume  of  a  Rectangular  Solid  =  Length  X  "Width  x 
Height. 

EXERCISES. 

1.  How  many  cubic  feet  in  a  pile  of  wood  4  ft.  high, 
8  ft.  long,  and  4  ft.  wide  ? 

Process.  Explanation. 

Base.  h.  v.  L  Volume   =  =   len§th    X 

f«  v  4^  v  4  —  S9  v  4  --  198  •        width  X  heiSht 

2.  .-.  The  volume  =  8  X 

4  X  4  =  128  (cubic  feet). 
The  factors  of  the  volume  must  be  expressed  in  the  same  denomination. 

2.  Find  the  volume  of  the  following  solids : 


Length. 

1.  20ft. 

Width. 
8ft. 

Height. 

Thickness. 
7ft, 

2.  10  in. 

9  in. 

12  in. 



3.  9  yd. 
4.  15ft, 
5.  9  ft.  6  in. 
6.  13  ft.  4  in. 
7.  11  ft.  8  in. 

6yd. 
11  ft. 
7ft. 
10ft. 
11  ft. 

10yd. 

10  ft. 
9ft. 

10ft. 

13ft. 

8.  13  ft.  3  in. 

12ft. 



9ft. 

9.  9  ft.  10  in. 

9  ft.  6  in. 



9ft. 

10.  25  ft.  9  in. 

25ft. 

100  ft. 

_____ 

BOARD  MEASURE  247 

3.  Find  the  value  of  the  cords  of  wood  in  the  follow- 
ing piles : 

Length.  Width.  Height.  Cost  per  cord. 

1.  35ft.  4ft.  ,8ft.  $3.00 

2.  43ft.  4ft.  16ft.  $3.25 

3.  59  ft.  27  ft.  13  ft.  $3.50 

4.  144  ft.  12  ft.  16  ft.  $4.00 

5.  200  ft.  25  ft.  25  ft.  $5.50 

4.  Find  the  cost  of  the  following  stone  piles : 

Height.  Length.  Thickness.     Cost  per  perch. 

1.  8ft.  50ft.  2ft.  $2.25 

A  perch  =  24.75cu.  ft. 

Suggestion:    *  X  60  Xjt  X  2.25 

2.  9ft.  75ft.  2J  ft-  $2.50 

3.  5  ft.  112  ft.  18  in.  $4.00 

4.  120  ft.  20  ft.  20  ft.  $3.00 

5.  30  ft.  8  ft.  9  in.     2  ft.  $  .75 

5.  Find  the  cost  of  excavating  the  following  cellars : 


Length. 

1.       40  ft. 

Width. 

20ft. 

Depth.      Cost  per  cubic  yd. 
8  ft.           30  cents. 

2.       30  ft. 

20ft. 

6  ft. 

25  cents. 

3.       38  ft. 

30ft. 

8ft. 

45  cents. 

4.       56J  ft. 
5.  40  ft.  6  in. 

40J  ft. 
30  ft.  9  in. 

8}  ft, 
8|ft. 

50  cents. 
40  cents. 

BOARD  MEASURE. 

1.  In  measuring  lumber,  a  board  1  inch  thick  or  less  is 
treated  as  a  mere  surface. 

2.  A  board  1  ft.  wide,  12  ft.  long,  and  1  inch  thick  is 
bought  or  sold  as  containing  12  ft.  board  measure. 


248  ELEMENTARY  ARITHMETIC 

3.  Thickness,    however,    becomes    a   factor    when   it 
exceeds  1  inch. 

FORMULA. 

Length  (ft.)  X  Width  (ft.)  X  Thickness  (in.)  =  Feet  Board 
Measure. 

PROBLEMS. 

1.  How  many  feet,  board  measure,  are  there  in  a  timber 
30  ft.  long,  7  in.  wide,  and  6  in.  thick  ? 

Process.  Explanation. 

15  1.  7in.  =^ft. 

8?  X  —  X  -  =  105  ft.  2-  "^  X  TV  =  the  number  of  board 

1%  feet,  thickness  1  inch. 

3.  ^a  X  TV  X  f  =  the  number  of 
board  feet,  thickness  6  in. 

4.  By  cancelling,  we  have  105  ft.,  board  measure. 

2.  Find  the  number  of  board  feet  in  the   following 
pieces  of  lumber : 

1.  16  ft.  by  8  in.  6.  15  ft.  by  12  in.  by  6  in. 

2.  15  ft.  by  9  in.  7.  40  ft.  by  10  in.  by  10  in. 

3.  12  ft.  by  10  in.          8.  24  ft.  by  9  in.  by  12  in. 

4.  10  ft.  by  12  in.          9.  40  ft.  by  10  in.  by  £  in. 

5.  14  ft.  by  18  in.        10.  12  ft.  by  6  in.  by  f  in. 

3.  Find  the  cost  of  a  dozen  boards,  each  15  ft.  long 
and  12  in.  wide,  at  $18  per  M.  feet. 

4.  I  have  30  joists  20  ft.  long,  18  in.  wide,  4  in.  thick. 
How  many  feet,  board  measure,  in  them  ?     Find  their 
cost  at  $20  per  M. 

5.  Find  the  weight  of  a  plank  15  ft.  9  in.  long,  10  in. 
wide,  and  2  in.  thick,  at  3^  Ib.  per  board  foot. 

6.  Find  the  cost  of  17  planks,  14£  ft.  long,  10  in.  wide, 
and  4  in.  thick,  at  $18^  per  M.,  board  measure. 


PERCENTAGE  249 

7.  How  many  board  feet  in  a  stick  of  timber  46  ft.  long, 
10  in.  thick,  12  in.  wide  at  one  end  and  9  in.  wide  at  tbe 
other  end  ? 

Suggestion :  Use  one-half  the  sum  of  the  end  widths. 


PERCENTAGE. 

The  decimal  fractions  of  most  general  importance  have 
for  their  denominator  100 ;  as,  yj^  =  .01,  yf^  =  .05,  -ffo 
-=.25. 

These  fractions  have  given  rise  to  Percentage,  which 
means  computation  by  hundredths. 

Per  Cent,  means  by  the  hundred ;  its  symbol  is  %. 

5  per  cent,  is  written  thus  :  5 %.     5%  =  yf^  =  -£$  =  .05. 

To  find  5%  of  $200  we  must  take  yjfo-  or  -£$  or  .05  of 
$200. 

yj^  of  $200  =  how  many  dollars  ? 
^  of  $200  ==  how  many  dollars  ? 
.05  of  $200  =  how  many  dollars  ? 

The  $10  is  called  the  Percentage. 

The  $200  is  called  the  Base. 

The  5%  is  called  the  Rate. 

$200  +  $10  =  $210  is  called  the  Amount. 

$200  —  $10  =  $190  is  called  the  Difference. 


PRINCIPLE. 

The  base  multiplied  by  the  rate  equals  the  Per- 
centage. 


250  ELEMENTARY  ARITHMETIC 

EXERCISES. 

1.  15  per  cent.  =  15%  ==  ^  ==  -fff  =  .15. 

Express  in  like  manner  10  per  cent,  in  four  different 
forms.  Also,  20  per  cent.,  25  per  cent,,  50  per  cent.,  75 
per  cent.,  100  per  cent.,  125  per  cent.,  250  per  cent. 

2.  33J  per  cent.  =  fjft  ==  flfft  =  £  =  .33fc. 

Treat  in  like  manner  12^  per  cent.,  6J  per  cent., 
per  cent,  37^  per  cent,  87-^-  per  cent.,  66f  per  cent, 
per  cent.,  2J-  per  cent,  7|-  per  cent,  2  per  cent,  16-|  per 
cent. 

3.  |  per  cent,  ==  yfcr  =  yfor  ==  .005. 

Show  in  like  manner  the  values  of  J  per  cent,  J  per 
cent,  -J  per  cent.,  -f  per  cent,  |-  per  cent 

•WRITTEN   EXERCISES   AND   PROBLEMS. 
1.  What  is  6%  of  $535.06? 

Process.  Explanation. 

$535.06  i-  ?>%  =  T«T  =  A  =  -06- 

Qg  2.  To  take  Q%  of  $535.06,  we  multiply  it  by  .06,  and 

obtain  for  the  percentage  $32^10  -j-  . 


$32.1036 

2.  What  is  75%  of  $800? 

Process.  Explanation. 


3  200  1.  76#  =  T7A  =  I  =  -75. 

2  of  $$P0  =  $600.  2.  By  cancellation  f  of  $800  =  3  times 

$200  =  $600. 
3.  .-.  76#  of  $800  is  $600. 

NOTE.—  In  your  processes  use  the  form  of  the  rate  that  will  give  you 
the  percentage  most  readily. 


PERCENTAGE  251 

3.  Find  the  percentage  from  the  following  bases  and 
rates : 

Base.  Rate.  Base.  Rate. 

1.  $600,  1%.  11.  $6.50,                  6%. 

2.  $700,  2%.  12.  $8.95,                 7%. 

3.  $800,  3%.  13.  $10.60,                8%. 

4.  $1000,  4%.  14.  $5.25,                10%. 

5.  $1250,  5%.  15.  $25.06,              12%. 

6.  275  sheep,  28%.  16.  225  pounds,     25%. 

7.  4380  bushels,  45%.  17.  3628  pounds,  39%. 

8.  10,000  barrels,  12J%.  18.  8250  apples,     88%. 

9.  1200  men,  1J%.  19.  4440  dollars,  100%. 
10.  320  horses,  6J%.  20.  1898  years,     33J%. 

4.  A  grocer  paid  $356  for  sugar  and  sold  it  at  a  gain  of 
15%.     What  was  his  gain  ? 

5.  I  bought  46  cords  of  wood,  at  $3.50  per  cord,  and 
sold  it  at  a  gain  of  25%.     What  was  my  whole  gain  ? 

6.  What  will  50  sheep  cost  if  75  are  worth  $375,  and 
for  how  much  must  they  be  sold  to  gain  12J%  ? 

7.  I  sold  500  bushels  of  corn  for  $200.     The  corn  cost 
me  25%  more.     How  much  did  I  lose  on  each  bushel? 

8.  A  man  bought  2165  bushels  of  wheat  for  $1515.50. 
A  part  receiving  damage,  he  is  willing  to  lose  10%.   Find 
the  selling  price. 

9.  A   miller   charges   5%    for  grinding.     How   many 
quarts  will  he  take  when  he  has  ground  35  bushels  ? 

10.  Ninety  per  cent,  of  a  class  of  60  pupils  are  pro- 
moted.    How  many  are  not  promoted  ? 

11.  A  girl  spelled  correctly  99%  of  200  words.     How 
many  did  she  miss  ? 


252  ELEMENTARY  ARITHMETIC 

12.  A  lot  is  sold  for  $1050,  of  which  is  20%  is  profit. 
What  did  the  lot  cost? 

13.  How  much  did  I  gain  on  a  house  for  which  I  paid 
$9870,  my  profit  being  3%. 

14.  A  piano,  marked  $750,  was  sold  for  cash  at  35% 
less.     "What  was  the  selling  price  ? 

15.  A  man  invested  12J%  of  $15,320  in  stocks.     How 
much  did  he  invest  ? 


INTEREST. 

1.  Interest  is  money  paid  for  the  use  of  money.     It  is 
computed  at  a  rate  per  cent,  of  the  money  used. 

2.  The  money  used  is  called  the  Principal  (Base). 

3.  The  Principal  X  Rate  =  One  year's  interest  (Per- 
centage). 

4.  Interest  for  a  greater  or  less  time  than  one  year  = 
Principal  x  Rate  X  Years,  or  fraction  of  a  year. 

What  is  6  %  of  $500  ?     Of  $1000  ? 

What  is  the  interest  of  $500  for  1  yr.,  at  6%  ?  For  2 
yrs.  ?  For  5  yrs.  ?  For  10  yrs.  ?  For  25  yrs.  ? 

What  is  the  interest  of  $1000  at  6%  for  |  yr.  ?  J  yr  ? 
f  yr.  ?  9  mo.  ?  10  mo.  ? 

Percentage,  or  one  year's  interest,  multiplied  hy  time, 
expressed  in  years,  gives  you  what  ? 

What  are  the  three  factors  of  interest? 


PRINCIPLE. 

The  factors  of  interest  are  principal,  rate,  and 
time  (years). 


INTEREST  253 

FORMULA. 
Int.  =  Prin.  X  R.  X  Yr. 

1.  Find  the  interest  of  $1200  for  2  yrs.,  at  5%. 

Process.  Explanation. 

$1200  X  -05  =  $60.  1-  *>%  of  $1200  =  $60,  the  int.  for  1  yr. 

~  fci  on  2.  $60  X  2  =  $120,  the  int.  for  2  yrs. 

3.  .-.  the  int.  of  $1200  at  5fc  for  2  yrs.  = 
$120. 

2.  What  is  the  interest  of: 

1.  $250  for  1  yr.  at  6%  ? 

2.  $300  for  1  yr.  at  5%  ? 

3.  $500  for  1  yr.  at  4%  ? 

4.  $600  for  1  yr.  at  3%  ? 

5.  $1000  for  lyr.  at7%? 

6.  $275  for  2  yrs.  at  8%  ? 

7.  $236  for  3  yrs.  at  5%  ? 

8.  $1673  for  4  yrs.  at  6%  ? 

9.  430.87  for  5  yrs.  at  7^? 

10.  $2846  for  6  yrs.  at  8%  ? 

11.  $12.46  for  H  yrs.  at  9%  ? 

12.  $126.37  for  1  yr.  at  1%  ? 

13.  $876  for  lyr.  at  S%  ? 

14.  $327  forf  yr.  at  6%  ? 

15.  $56.79  for  5£  yrs.  at  4%  ? 

16.  $182.34  for  2|  yrs.  at  6%  ? 

17.  $182.34  for  2  yrs.  6  mo.  at  6%  ? 

18.  $432.81  for  1  yr.  9  mo.  at  5%  ? 

19.  $3843  for  3  yrs.  3  mo.  at  7%  ? 

20.  $129.95  for  1  yr.  8  mo.  at  10%  ? 


254  ELEMENTAEY  AEITHMETIC 

3.  Find  the  interest  of  $530.60  for  6  yr.  8  mo.  at  6%. 

Process.  Explanation. 

2  1.  &%  =  T^.     6  yr.  8  mo.  = 

106.12  *  80mo 

X  g  X  W        (fcoin  94  r^u/i-       ^        •     •     iv 

x  100  --  ==  W^l^'^4.  2.  Multiplying  the  principal  by 

ft  -j-^,  the  rate,  we  get  the  interest 

for  1  year.     By  dividing  the  inter- 

est for  1  year  by  12,  we  get  the  interest  for  1  mo.  Then  multiplying  the 
interest  for  1  mo.  by  80,  the  number  of  months,  we  get  the  interest 
required.  The  process  is  shortened  by  cancellation. 

1.  $523.30  for  3  yrs.  9  mo.  at  5%. 

2.  $429.47  for  4  yrs.  7  mos.  at  7%. 

3.  $536.35  for  5  yrs.  6  mos.  at  6%. 

4.  $796.70  for  4  yrs.  8  mos.  at  8%. 

5.  $548.29  for  6  yrs.  3  mos.  at  9%. 

6.  $415.38  for  5  yrs.  5  mos.  at  4%. 

7.  $847.27  for  3  yrs.  10  mos.  at  7%. 

8.  $1046.60  for  2  yrs.  3  mos.  at  6%. 

9.  $858.94  for  3  yrs.  5  mos.  at  6%. 
10.  $1072.70  for  1  yr.  11  mos.  at  6%. 

4.  Find  the  interest  of  $635.48  for  3  yr.  6  mo.  18  da. 

at  4%. 

Process. 
6.3548  14.2 

=  6'3548  X  142'  = 


Explanation. 

1.  4^  =  yf  o-     3  yr.  6  mo.  18  da.  =  42.6  mo. 

2.  The  principal  multiplied  by  the  rate,  ¥^,  gives  the  interest  for 
1  yr.  ;  the  interest  for  1  yr.,  divided  by  12,  gives  the  interest  for  1  mo.  ; 
the  interest  for  1  mo.  multiplied  by  the  number  of  months  gives  the 
interest  for  full  time.     By  cancellation  we  find  $90.24  to  be  the  required 
interest. 


GENERAL  REVIEW  255 

5.  Find  the  interest  of  $480  for  2  yr.  11  mo.  10  da. 

at  5%. 

Process.  Explanation. 

#  212  l-  ^%  =  dhr-     2  vr-  n 

g?0  X  ff  X  ?000  _  JMMJ.  __  $70  £7  mo.  10  da.  =  1060  da. 

W  X  m        ^  2.  Finding  the  interest  for 

nrt.  -rq 

one  year,  as  before,  we  then 
divide  the  interest  for  one  year 

by  360,  and  thus  get  the  interest  for  1  day.     The  interest  for  1  da.  multi- 
plied by  the  number  of  days  gives  the  interest  for  fiHl  time. 

The  Principal  -f  the  Int.  =  the  Amount.     $480  +  $70.67  =  $550.67. 

6.  Find  the  interest  and*the  amount  of: 

1.  $470.35  for  3  yr.  8  mo.  18  da.  at  6%. 

2.  $516.50  for  4  yr.  6  mo.  1.5  da.  at  7%. 

3.  $318.47  for  3  yr.  7  mo.  10  da.  at  5%. 

4.  $534.46  for  4  yr.  9  mo.  27  da.  at  7%. 

5.  $830.27  for  4  yr.  8  mo.  17  da.  at  6%. 

6.  $579.47  for  4  yr.  10  mo.  12  da,  at  6%. 

7.  $320.58  for  3  yr.  6  mo.  13  da.  at  7%. 

8.  $436.45  for  2  yr.  2  mo.  24  da.  at  8%. 

9.  $547.44  for  5  yr.  3  mo.  19  da.  at  4%. 
10.  $308.56  for  6  yr.  4  mo.  9  da.  at  3J%. 

GENERAL    REVIEW. 
1.  Add: 

1.  $295,746.97J  2.  $374,116.96  3.  $6,604.00 

36,905.73  49,573.88  56,948.35 

9,867.96  260,087.61  35,439.50 

999.68  60,429.03  678,950.70 

5,437.99J  47,596.84  30,597.05 

69,141.16  10,970.55  65,300.40 

309,609.78  87,046.00  3,689.74 

48,765.99  9,900.67  439.00 


256  ELEMENTARY  ARITHMETIC 

2.  Find  the  sum  of  .37,  6.3,  .009,  10.74,  1.07,  58.93, 
748.57  and  2.034. 

3.  Find  the  value  of.a^ftfr  +  9T3¥  +  608  -f  39TVg-  + 
^o  +  3.1416  -f  231. 

4.  Add  25  days  7  hours  51  minutes  and  26  days  28 
hours  and  21  minutes. 

5.  Find  the  value  of  3.5  bu.  3  pk.  -f-  35  bu.  3  pk.  -f- 
35  bu.  3  pk. 

6.  Add  24  Ib.  7  oz.,  6  Ib.  10  oz.,  36  Ib.  1 1  oz.,  5  Ib.  8  oz. 

7.  Add  12  rd.  4  yd.  2  ft.,  5  rd.  2  ft.,  5  yd.  1  ft,  7  rd. 

8.  Find  the  sum  of  60^-  +  49^  +  18^  +  6}  +  901 

9.  Find  the  sum  of  f  -f-  f  -f  f  -f  5f . 

1O.  Add  234  cu.  yd.  18  cu.  ft.  566  cu.  in.,  149  cu.  yd.  9 
cu.  ft.  19  cu.  in.,  198  cu.  yd.  11  cu.  ft.  1000  cu.  in.,  70  cu. 
yd.  23  cu.  ft.  1267  cu.  in. 


(1.) 

11.  From  9932563 
take  7953240 

(2.) 
From  6875396 
take  5927387 

(3.) 

From  8735009 
take  7295394 

12.  Find  the  value  of: 

(i.)  (2.)  (3.)  (4.) 


(5.)  (6.) 

7320.00  —  6837.13.  4197.23  —  2076.88. 

(7.)  (8.) 

173.09  —  134.137.  8394.46  —  4153.76. 

13.  From   five  hundred   eighty  and  sixty-seven  ten- 
thousandths  take  ninety-six  and  forty-nine  millionths. 

14.  George  Washington  was  born  Feb.  22,  1732.    How 
old  was  he  July  4,  1776  ? 


GENERAL  REVIEW  257 

15.  From  16  rd.  take  4  rd.  1  yd.  1  ft. 

16.  From  7  bu.  3  pk.  6  qt.  take  6  bu.  2  pk.  7  qt. 

17.  How  much  is  left  if  you  take  4  gal.  3  qt.  1  pt.  from 
a  milk-can  containing  10  gal.  ? 

18.  Subtract  59.9078  from  64.08. 

19.  Find  the  value  of  (28  —  9)  —  (27  —  25)? 

20.  A  man  bought  a  factory  for  $27,000,  and  sold  it  for 
$25,310.625.     How  much  did  he  lose  ? 


21.  Multiply:  39,086  by  3049 ;  27,389  by  8375 ;  43,009 
by  3468. 

22.  Find  the  value  of  5£  X  7£;  16^  X  1$;  7T3<r  X  9£; 
3.1J  X  2&. 

23.  Find  the  value  of:    .07646  X  76;    10.025  X  7.29; 
.036  X  -94J. 

24.  What  cost  87  yd.  of  muslin  at  16f  cents  per  yard? 

25.  What  is  the  value  of  2758  bu.  of  wheat  at  $.70J 
per  bushel? 

26.  Multiply  4  mi.  130  rd.  11  ft.  by  38. 

27.  What  is  the  value  of  .875  A.  of  land  at  $.33J  per 
square  rod  ? 

28.  Find  the  value  of  1|  X  TV  of  f  of  £  of  -4**.. 

29.  Find  the  value  of  (f  of  $)  +  (f  of  f )  —  (J.  of  2). 

30.  Multiply  10  hhd.  20  gal.  3  pt.  by  9. 


31.  Divide  62,098,347  by  5009;  87,329,046  by  8920. 

32.  Divide  -^  by  £;  17|f  by  |l;  7|  by  3T37;   7  by  12|. 

33.  Divide  .504  by  .024;  123.6  by  .01;  829.31  by  .019. 

17 


258  ELEMENTAEY  AEITHMETIC 

34.  Divide  26  mi.  28  rd.  14J  ft.  by  7 ;  74  cd.  19  cu.  ft. 
by  9. 

35.  Find  the  value  of  2J  x  (f  -*-  J)  X  |. 

36.  Multiply  3^  by  17f,  and  divide  the  result  by  2J 
of  2§. 

37.  If  a  barrel  of  flour  is  worth  $6f ,  how  many  barrels 
will  $510  buy? 

38.  If  24  men  do  a  piece  of  work  in  105  days,  how 
long  will  it  take  72  men  to  do  it  ? 

39.  At  18f  cents  per  C.,  how  many  laths  can  be  bought 
for  $37.50. 

40.  Divide  f  of  £  of  £  of  f  by  £  of  f  of  •&  of  4. 


41.  Find  by  cancellation  the  value  of  ^  *5  *of  *  J>  ^ 

«'  X  '  X  ^l  X  ou 

144  X  17  X  45  X  52  .  57  X  119  X  16 
13  X  9  X  34  X  12    '    17  X  12  X  19' 

42.  How  many  dresses,  each  containing  15  yd.,  can  be 
made  from  25  pieces  of  cloth,  each  containing  45  yd.  ? 

43.  Reduce  to  lowest  terms  :  ff  ,  fM*»  iHi»  W&- 

44.  Find  the  prime  factors  of  295,  556,  648,  1063,  1495. 

45.  Reduce  to  higher  terms  :  f  to  27ths,  ff  to  819ths, 
^  to  lOOOths. 

46.  Reduce  to  whole  or  mixed  numbers: 


47.  Reduce  to  improper  fractions  :  68f  ,  400^, 


48.  Reduce  4  bbl.  5  gal.  1  pt.  to  gills;  5  hhd.  7^  gal. 
to  pints. 

49.  Reduce    to    higher   denominations  :    845,356    in.  ; 
1,000,000  cu.  ft;  7016  dr. 


GENERAL  REVIEW  259 

5O.  Reduce  to  lower  denominations :   J-  T. ;   .75  bu. ; 
.81  sq.  mi. ;  .385  da. 


51.  Find  the  value  of  (4f  +  6£  —  3f)  ~-  |  +  (3J  — 
2i)  X  6. 

52.  Find  the   value   of  81.8   +   35.625  —  38.875  — 
2.0034. 

53.  What  is  the  sum  of  303  thousandths,  4108  rnil- 
lionths,  635  ten-thousandths,  803  ten-millionths  ? 

54.  Multiply  .8745  by  100;  by  1000;  by  100,000;  by 
1,000,000. 

55.  Divide  .8745  by  100;   by  1000;  by  100,000;  by 
1,000,000. 

56.  Divide  .08  by  1.611;   40,000  by  .00004;  144  by 
1728. 

57.  How  many  barrels  of  apples,  at  $2.25  per  barrel, 
can  be  bought  for  $29£  ? 

58.  If  illuminating  gas  is  sold  at  the  rate  of  $1.00  per 
M.  cubic  feet,  how  much  will  52,437  cu.  ft.  cost  ? 

59.  If  a  railroad  train  runs  35.75  miles  per  hour,  in 
how  many  hours  will  it  run  143  miles  ? 

60.  A  farmer  sold  21J  dozen  eggs  at  $.18f  a  dozen, 
and  bought  14 J  yd.  of  cloth  at  12J  cents  a  yard.     How 
much  money  had  he  left  ? 


61.  How  many  acres  in  a  rectangular  field  70  rd.  wide 
and  90  rd.  long  ? 

62.  The  length,  breadth,  and  thickness  of  a  solid  are, 
respectively,  9  ft.  4  in.,  10  ft.  6  in.,  and  7  ft.  8  in.     Find 
its  volume. 


260  ELEMENTARY  ARITHMETIC 

63.  How  many  cords  of  wood  in  a  pile  4  rd.  long,  2| 
yd.  high,  and  4  ft.  wide  ? 

64.  How  many  cubic  inches  in  a  cubic  rod  ? 

65.  How  many  board  feet  in  38  planks  10  in.  wide,  16 
ft.  long,  and  2  in.  thick  ? 

66.  How  many  acres  in  a  rectangle  24 \  rd.  long  by 
16.02  rd.  wide? 

67.  How  many  square  feet  in  the  4  sides  of  a  room  21 J 
ft.  long,  16J  ft.  wide,  and  13  feet  high? 

68.  What  is  the  cost  of  a  field  173  rd.  long  and  84  rd. 
wide  at  $35.75  per  acre  ? 

69.  How  many  board  feet  in  a  3-inch  plank  18  ft.  long 
and  14  in.  wide  ? 

70.  How  much  will  a  stick  of  timber  40  ft.  9  in.  long, 
1  ft.  3  in.  wide,  and  1  ft.  9  in.  thick  cost,  at  25  cents  a 
cubic  foot  ? 


71.  Find66f%  of  $314.16. 

72.  Find  the  interest  of  $57.35  for  90  da.,  at  6%. 

73.  Find  the  amount  of  $691.75  for  15  da.,  at  6%. 

74.  A  money-lender  loaned  $840.50  for  2  yr.  and  8 
mo.,  at  5J%.     What  amount  of  money  was  due  him  at 
the  end  of  the  time  ? 

75.  I  loaned  a  friend  $460  on  Jan.  1,  1898,  at  6%  in- 
terest.    If  the  loan  is  not  repaid  till  Jan.  1,  1900,  how 
much  money  will  then  be  due  me  ? 

76.  Find  the  amount  of  $108.46  for  4  yr.  8  mo.  5  da., 
at  10%. 

77.  Find  the  interest  and  amount  of  $500  for  2  yr.  2 
mo.  2  da.,  at  6%. 


GENERAL  REVIEW  261 

78.  Find  the  interest  of  $1000  for  30  days.     For  33 
days. 

79.  $  100,  put  on  interest  Dec.  1,  1898,  will  amount  to 
how  much  money  on  Dec.  31, 1899,  at  5%,  reckoning  365 
days  in  a  year  ? 

80.  I  bought  a  horse  for  $90.00.     I  sold  him  at  a  gain 
of  25%.    I  loaned  the  money  thus  received,  for  a  year,  at 
7%.     How  much  money  was  due  me  at  the  end  of  the 
year? 


81.  Two  men  start  at  the  same  time  and  place  and 
travel  in  opposite  directions,  one  at  the  rate  of  35  mi.  per 
day,  the  other  at  the  rate  of  42  mi.  per  day.     How  far 
will  they  be  apart  at  the  end  of  19  days  ? 

82.  A.,  B.  and  C.  have  together  $209.     A.  and  C.  have 
$155 ;  A.  and  B.  have  $109.     How  much  has  each  ? 

83.  The  greater  of  two  numbers  is  5067,  and  their 
difference  is  4760.     What  is  the  less  number  ? 

84.  The  divisor  is  645  and  the  quotient  43.     What  is 
the  dividend  ? 

85.  The  subtrahend   is  45,304  and  the  remainder  is 
9807.     What  is  the  minuend  ? 

86.  A  wagon  wheel  is  16  ft.  8  in.  in  circumference. 
How  many  revolutions  will  it  make  in  going  5  miles  ? 

87.  If  f  of  an  acre  of  land  produces  120  bushels  of 
potatoes,  how  many  bushels  will  4J  acres  produce  ? 

88.  A  man  who  owned  4  of  a  mill  sold  4-  of  his  share 

o  O 

for  $2500.     What  was  the  value  of  the  mill  at  that  rate, 
and  what  the  value  of  the  man's  share  ? 

89.  James  can  do  a  piece  of  work  in  12  days,  and 


262  ELEMENTARY  ARITHMETIC 

Robert  can  do  it  in  10  days.     In  what  time  can  they  both 
do  it? 

9O.  Three  children  inherited  in  equal  shares  a  farm  of 
473  A.  112  sq.  rd.     How  much  land  did  each  receive? 


91.  A  vintner  has  168  gal.  3  qt  1  pt.  of  wine.     How 
much  is  it  worth  at  $2.62J  per  gallon  ? 

92.  4  T.   17  cwt.  75  Ib.   of  hay  being  sold  brought 
$10.75  per  ton.     What  was  the  total  receipt  for  the  hay  ? 

93.  How  many  dresses  containing  8  yd.  2  ft.  each  can 
be  made  out  of  390  yd.  of  cloth  ? 

94.  A  lot  contains  12  A.     It  is  30  rd.  wide.     How 
long  is  it? 

95.  A.  has  J  as  much  money  as  B.     They  both  have 
$581.     How  much  has  each  ? 

96.  A  room  measures  on  the  floor  20  ft.  by  18  ft. 
How  much  carpet  will  cover  it,  and  what  will  the  carpet 
cost  at  $l.l2J  per  square  yard  ? 

97.  I  paid  a  debt  of  $295.85,  which  had  been  upon 
interest  11  mo.  25  da.  at  7%.     What  amount  did  I  pay? 

98.  What  number  divided  by  If  will  give  a  quotient 
of  4f  ? 

99.  A   merchant   bought    1250    barrels   of  flour    for 
$6250.     At  what  price  per  barrel  must  he  sell  it  to  make 
a  profit  of  12J%? 

1.  Percentage  =  Base  X  Rate. 

2.  Interest  =  Percentage  x  Time. 

3.  Interest  =  Base  -X  Kate  x  Time. 

100.  Base  =  ?  [1  and  3.]  Rate  =  ?  [1  and  3.]   Time  =  ? 
[2  and  3.] 


ANSWERS. 


ADDITION. 

9.  26  feet. 

10.  40  inches. 

Page  7O. 
2.  1.  24.          15.  23. 

29.  28. 

11.  17  boys  ;  11  girls  ;  28  pupils. 
12.  16  units  ;  36  cents. 

2.  22.          16.  17. 

30.  28. 

3.  24.          17.  22. 

31.  35. 

Page  73. 

4.  19.          18.  23. 

32.  21. 

13.  33  lines.         14.  45.         15.  35. 

5.  28.          19.  17. 

33.  26. 

6.  20.          20.  21. 

34.  30. 

Page  74. 

7.  21.          21.  28. 

35.  23. 

2.  1.  1368.       4.  1760.       7.  492. 

8.  21.          22.  28. 

36.  24. 

2.  1125.       5.  1652.       8.  589. 

9.  24.         23.  27. 

37.  29. 

3.   1531.       6.  688. 

10.  24.          24.  24. 

38.  27. 

11.  23.          25.  25. 

39.  29. 

Page  75. 

12.  25.          26.  25. 

40.  28. 

9.  $708.                  12.  $137.44. 

13.  27.          27.  35. 

10.  $38.23.                13.  $175.81. 

14.  15.          28.  33. 

11.  $225.44.              14.  $2874.42. 

Page  71. 

3.  1.  90309.              3.  9055. 

3.  1.  29.           8.  32. 

15.  31. 

2.  10970.             4.  5293. 

2.  33.            9.  29. 

16.  28. 

4.  26765.                  9.  243076. 

3.  29.          10.  28. 

17.  30. 

5    13593.                10.  $278.85. 

4.  32.          11.  39. 

18.  41. 

6.  22070.                 11.  $3261.69. 

5.  30.          12.  29. 

19.  31. 

7.  8110.                   12.  $1398.93. 

6.  34.          13.  27. 

20.  36. 

8.  135929. 

7.  37.          14.  37. 

1.  18.                         3.  24 

feet. 

Page  76. 

2.  12  years.               4.  33 

books. 

13.  $342.809.            15.  $710.688. 

14.  $833.783. 

Page  72. 

16.  1.  56065.             6.  547000000. 

5.  15  pieces. 

2.  15071.             7.  29000788. 

6.  13  gills. 

3.  24004.              8.  29900. 

7.  11  pounds. 

4.  435000.            9.  9006439. 

8.  18  problems. 

5.  5004867.        10.  21847. 

263 

264 


ANSWERS 


17.  1.  $1160.33.         3.  $703.78. 

1.  702. 

4.  3420.         7.  101.20. 

2.  $382.99. 

2.  1.00. 

5.  .          8.  252.15. 

Page  77. 

3.  2.50. 

6.  .          9.  3131. 

4.  $1638.02.         5.  $1500.3725. 

1.  $16992.       3.  $800.        5.  $8818. 
2.  47624.         4.  21.00.       6.  1164. 

Page  86. 

10.  1000. 

15.  1732. 

Page  78. 

11.  2030. 

16.  401. 

7.  $126.90;  217. 

12.  5504. 

17.  . 

8.  2,622,254. 

13.  1115. 

18.  202.20. 

9.  $1796. 

14.  3632. 

19.  106. 

10.  $9265. 

11.  842. 

12.  16906. 
13.  $8360. 

Page  89. 

14.  817;  $8.17;  126. 

3.  1.  169. 

16.  76. 

2.  212. 

17.  20.39. 

Page  79. 

3.  391. 

18.  7.92. 

15.  1.  13701. 

4.  329. 

19.  16.48. 

2.  12248. 

6.  264. 

20.  21.05. 

3.  135667. 

6.  387. 

21.  9.25. 

4.  128257. 

7.  115. 

22.  35.38. 

5.  1XDCCCXCVIII  =  9898. 

8.  32. 

23.  35.91. 

9.  66. 

24.  28.41. 

SUBTRACTION. 

10.  509. 

25.  449.48. 

Page  84. 

11.  82. 

26.  269.89. 

2.  1.  211.         8.  341.         15.  1343. 

12.  108. 

27.  158.07. 

2.  552.         9.  210.        16.  4200. 

13.  46. 

28.  688.89. 

3.  341.        10.  400.        17.  3213. 

14.  567. 

29.  2578.99. 

4.  430.        11.  351.        18.  3541. 

15.  167. 

5.  602.        12.  532.        19.  3233. 

6.  201.        13.  440.        20.  7111. 

7.  633.        14.  311. 

Page  9O. 

Page  85. 

1.  943. 

8.  1096. 

21.  $22.11.            27.  61.12. 

2.  2031. 

9.  1. 

22.  24.20.              28.  71.000. 

3.  1435. 

10.  999000. 

23.  16.13.              29.  45.90. 

4.  1206. 

11.  593. 

24.  13.45.               30.  10.426. 

5.  35433. 

12.  125.77. 

25.  36.14.              81.  610.17. 

6.  1090. 

13.  146.882. 

26.  53.35. 

7.  77655. 

14.  92760000. 

ANSWERS 


265 


5.  1.  391.       11.  34371. 

Page  96. 

2.  417.       12.  540100. 

13.  A.  450  ;  B.  625  ;  C.  961. 

3.  387.       13.  459890. 

14.  0. 

4.  51.        14.  5170174. 

15.  $11251. 

5.  1991.      15.  136353. 

16.  1578620818. 

6.  4886.       16.  1228608. 

17.  87. 

7.  1603.      17.  132187. 

18.  $1056.75. 

8.  1962.      18.  4745334. 

19.  217. 

9.  2218.       19.  307139. 

20.  Summer,  2  days. 

10.  54029.     20.  1979323. 

21.  $87.28. 
Page  97. 

1.  7202.         3.  52581. 
2.  951691. 
Page  91. 

22.  Mary,  5  cts. 
23.  $3344;  $4162.84. 

24.  24;  34;  6.684. 

4.  6462.        10.  298,470,000. 
5.  11020.       11.  10,179,000; 

25.  $.50;  $.375;  $.625. 
26.  936085. 

6.  6757.            19642000. 

27.  1216738. 

7.  332100.      12.  $204.96. 
8.  999891.      13.  46.8. 

28.  $3.00. 
29.  DCXXXI. 

9.  976000000.    14.  1128. 

o/\   /~1 

30.  C. 

Page  93. 

31.  $9850. 
32.  $7240;  $2760. 

1.  $7.     5.  17.     9.  $19.50. 

2.  10.     6.  $12.    10.  1801. 

MULTIPLICATION. 

3.  9.      7.  65. 

Page  103. 

4.  15.     8.  $2. 

1.  1735.       18.  184344. 

Page  94. 

2.  2192.       19.  380736. 
3.  2120.       20.  376096. 

11.  0.     13.  28.     15.  $3.00. 

4.  3748.       21.  395586. 

12.  18.     14.  $80. 

5.  1740.       22.  253664. 

2.  1.  10654.      6.  885. 

6.  2715.       23.  340784. 

2.  2185.       7.  30443. 

7.  3420.       24.  213409. 

3.  2367.       8.  58852. 

8.  4715.       25.  246912. 

4.  50895.      9.  2517. 

9.  12288.      26.  2367036. 

5.  11388.     10.  487783. 

10.  41559.      27.  1382712. 

11.  53670.      28.  4506170. 

Page  95. 

12.  42266.      29.  3407340. 

3.  7202.        8.  $16.75. 

13.  71856.      30.  864192. 

4.  $14750.       9.  94. 

14.  50768.      31.  7120984. 

5.  $16675.      10.  $8454.20. 

15.  48564.      32.  4111101. 

6.  $2693.23.     11.  $2.10. 

16.  21980.      33.  4382715. 

7.  33064.       12.  $22.735. 

17.  482805. 

266 


ANSWEKS 


Page  104. 

Page  108. 

1.  31.50.       5.  2920. 

3.  1.  15910.     26.  4694.18. 

2.  200.        6.  10080. 

2.  20860.     27.  2374.40. 

3.  2.50.        7.  15052.94. 

3.  31680.     28.  1675.28. 

4.  864.        8.  31680. 

4.  45144.     29.  1493.750. 

9.  1.  56.     6.  55.     9.  54. 

5.  59031.     30.  4457.05. 

2.  15.     6.  16.     10.  0. 

3    Of)            7   QO 

6.  43148.     31.  7545020. 
7.  32795.     32.  19924450. 

.  UZ.        f.  Jo. 

8.  61050.     33.  17662020. 

4.  38.     8.  83. 

9.  39220.     34.  21510354. 

10.  4177182. 

10.  43601.     35.  10374925. 

1.  4374762.    1.  363655. 

11.  56576.     36.  20312880. 

2.  5827927.    2.  8169459. 

12.  63640.     37.  21558420. 

3.  4509248.    3.  1603774. 

13.  620412,    38.  19070156. 

4.  5620734.    4.  1189087. 

14.  836060.    39.  9979902. 

6.  3755255.    5.  1810659. 

15.  914620.    40.  20640080. 

6.  4507712.    6.  1439615. 

16.  237303,    41.  1236985904. 

7.  2399868.    7.  4150327. 

17.  266350.    42.  2067421020. 

18.  237728.    43.  4133249372. 

19.  178365.    44.  3515881096. 

20.  386496.    45.  430509170. 

Page  1O6. 

21.  2160.16.    46.  7747702200. 

5.  1.  67900.      6.  450000. 

22.  2129.40.    47.  6903130377. 

2.  480000.     7.  4000000. 

23.  1411.60.    48.  8757235116. 

3.  26900.      8.  5700000. 

24.  1914.84.    49.  1679687810. 

4.  3760000.    9.  77900. 

25.  2591.60.    50.  1919163400. 

6.  340500.    10.  14850. 

51.  3963930272. 

52.  12912828391. 

53.  17072262240. 

54.  5755196745. 

Page  1O7. 

55.  11825571472. 

11.  22800.     21.  227200. 

56.  7567934076. 

12.  23100.     22.  46100. 

57.  221461306500, 

13.  54320.     23.  9876000. 

58.  97406784000. 

14.  182100.    24.  93520. 

59.  207438609862. 

15.  203200.    25.  24048000. 

60.  70109427840. 

16.  543200.    26.  3288000. 

17.  171600.    27.  7952000. 

Page  JO9. 

18.  653800.    28.  4736000. 

1.  85064.        4.  1470. 

19.  3072000.    29.  10728000. 

2.  7285.        5.  110048, 

20.  4690000.   30.  994000. 

3.  698720.       6.  31710, 

ANSWERS 


267 


7.  4379.43.      11.  651222. 

Q.   R. 

Q.   R. 

8.  8565.78.      12.  614680. 

4.  1.  368  1. 

15.  1228  6. 

9.  89760.       13.  2243160. 

2.  245  1. 

16.  1406  5. 

10.  8883.        14.  $12000. 

3.  240  3. 

17.  1058  2. 

4.  183  1. 

18.  985   3. 

Page  11O. 

5.  137  2. 

19.  24371  1. 

15.  $878.85. 

6.  133  6. 

20.  17914  0. 

7.  124  3. 

21.  16472  1. 

Page  in. 

8.  107  2. 

22.  9479  2. 

1.  24791.        2.  175. 

9.  84   7. 

23.  9618  1. 

10.  675  5. 

24.  8256  2. 

Page  112. 

11.  1220  7. 

25.  13367  1. 

3.  600.         10.  424. 

12.  1979  1. 

26.  12084  2. 

4.  3125.20.      11.  999000. 

13.  1099  3. 

27.  6387  4. 

5.  75881.        12.  99900000. 

14.  3189  1. 

6.  720.         13.  5820. 

7.  $2140920.     14.  196.80. 

8.  7883.        15.  35. 

9.  $72035.       16.  1440. 

Page 

121. 

Q.   R. 

Q.   R. 

Page  113. 

28.  197  1. 

45.  1035  8. 

17.  1806.        21.  5219. 

29.  154  3. 

46.  1587  2. 

18.  58.24.       22.  636  profit. 

30  137  6. 

47.  1624  2. 

19.  12960.       23.  12  profit. 

81.  248  3. 

48.  1158  4. 

20.  374.        24.  2400. 

32.  71   6. 

49.  245   4. 

25.  1.  600. 

33.  382  1. 

50.  12865  1. 

34.  124  6. 

51.  19739  1. 

Page  114. 

35.  192  3. 

52.  13796  3. 

2.  698.       4.  $1798.98. 

36.  107  2. 

53.  11837  5. 

3.  79962. 

37.  239  1. 

54.  7765  1. 

38.  162  2. 

55.  28780  0. 

Page  12O. 

39.  532  4. 

56.  22669  2. 

2.  1.  321.  11.  1225.  21.  $5.54. 

40.  529  4. 

57.  16560  5. 

2.  321.  12.  1084.  22.  $13.53. 

41.  1206  3. 

58.  31240  0. 

3.  918.  13.  1218.  23.  $5.48. 

42.  945  2. 

59.  11853  7. 

4.  541.  14.  1366.  24.  $12.34. 

43.  1247  2. 

60.  10660  2. 

5.  307.  15.  496.   25.  $9.12. 

44.  1404  4. 

6.  647.  16.  597.   26.  $13.89. 

7.  315.   17.  1545.  27.  $9.08. 

1.  200. 

5.  65654. 

8.  337.   18.  984.   28.  $22.96. 

2.  176. 

6.  62256. 

9.  384.   19.  406.   29.  $8.88. 

3.  285. 

7.  66365 

10.  493.  20.  876.   30.  $3.91. 

4.  21120. 

268 


ANSWERS 


Page  122. 

8.  5f.  13.  $7351.   18.  248. 

9.  162;  126.  14.  1120.      19.  1342. 

10.  588.  15.  4840.      20.  180. 

11.  144  hr.       16.  52. 

12.  4688.         17.  6542. 


Page  125. 


4.  1.  8TV 
2. 
3. 
4. 


16.  32/AV 
17. 


6. 
7. 
8. 
9. 
10. 

11.  198^. 

12.  61f§. 

13.  16HJ. 


20.  llflfof. 

21.  12ttft. 

22.  llJHfr. 

23.  123TW<r. 
24. 

25. 
26. 
27. 
28. 
29. 
30. 


Page  127. 


5.  1.  789. 

2.  482TV 

3.  305TV 

4.  583^1. 
6.  292TV 

6.  605|f 

7.  154H. 

8.  201T37. 

9.  144TV 

10.  203TV 

11.  221. 

12.  21926T. 

13.  209. 

14.  135£f. 


16.  135f§. 

16.  sm. 

17.  66|f 

18.  537^. 

19.  41|f. 

20.  74^. 

21.  1127V 

22.  53f|. 

23.  83f  *. 

24.  737f. 

25.  128ff 

26.  221. 

27.  219£f. 

28.  209. 


29.  135|f. 

30.  159|7. 

31.  1052. 

32.  994.V 

33.  862if 

34.  2108T%. 

35.  1291TV 

36.  3179^. 

37.  577|f. 

38.  547^. 

39.  846ff. 

40.  495^. 

41.  552f£. 

42.  Sllfo. 

43.  144^. 
44. 


45. 

46. 

47. 

48. 

49.  206. 

50. 

51. 

52.  316ifo. 

63.  443i|f. 

54.  27T<&. 

55. 

56. 

57. 

58. 

69. 

60.  245&\. 


Page  128. 


6.  1.  917fff  • 
2.  1452^?. 


17. 

18. 
19. 
20. 

21.  246f|7f. 

22.  1594f|f£. 

23.  lOOlffff. 

24.  262/77/7. 

25.  1030|fff 
26. 

27. 
28. 

29.  935f|ff. 

30.  679|JJJ 

32.  452|||fS 


Page  129. 

1.  115.  3.  135.  5."60T^7. 

2.  25.  4.  135.  6.  16. 


4.  3897|f. 

5.  1375^. 

6.  463&V 

7.  801|ff 

8.  397/7V 

10.  860|||. 

11. 
12. 
13. 
14. 
15. 
16. 


ANSWERS 


269 


Page  ISO. 

Page  14O. 

7.  14. 
8.  4732. 

12.  539.          17.  280. 
13.  135.           18.  25. 

2.  1.  2,  2,  3,  19. 
2.  2,  2,  3,  3,  3,  3. 

9.  2093. 

14.  21890.       19.  2. 

3.  2,  2,  2,  2,  2,  7. 

10.  130. 

15.  329. 

4.  2,  2,  2,  43. 

11.  494}jf 

.     16.  480. 

5.  2,  2,  2,  2,  3,  3. 

• 

Page  131. 

6.  3,  3,  6,  5. 

20.  75. 

22.  3|^.        24.  36. 

7.  2,  2,  199. 

21.  16f^f 

.    23.  97. 

8.  2,  2,  2,  2,  2,  2,  3,  3. 

9.  2,  2,  2,  2,  2,  2,  2,  2. 

1.  228. 

4.  147.            7.  180. 

10.  2,  2,  3,  5,  5. 

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Page  135. 

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41.  784. 

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42.  138. 

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270 


ANSWERS 


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272 


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ANSWERS 


273 


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274 

ANSWERS 

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ANSWERS 


275 


Page  2O3. 

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6.  91b.                   12.  $1.269. 

16.  2.5. 

7.  188  T.                13.  12  cd. 

17.  36.4. 

8.  50.385  -f  T.      14.  18. 

18.  43.2. 
19.  82160. 
20.  12345. 

15.  1.  $.50.                4.  $21.33f. 
2.  4cts.               5.  $.87|. 
3.  90cts. 

21.  .32. 

22.  960. 

23.  16.68^. 

Page  21O. 

24.  .204. 

6.  $.05.                9.  $.08. 

25.  .0625. 

7.  $.175.            10.  $5.35. 

26.  250. 

8.  $.06J. 

27.  3.27. 

16.  1.  .48.                 6.  27. 

28.  5.16. 

2.  .00301.            7.  1. 

29.  1.3. 

3.  72.63.              S.  .06. 

30.  88.90. 

4.  1.70.                9.  69.917. 

31.  5.27. 

6.  6.03.              10.  646.' 

276 


ANSWERS 


Page  214. 

Page  232. 

1.  $2.25.                      7.  $52.827. 

9.  96  oz.                  18.  2592  cu.  in. 

2.  $778.615.               8.  $.15. 

10.  20  pk.                 19.  6  pwt. 

3.  $63.41.                   9.  $28. 

11.  24  pt.                  20.  6  Ib. 

4.  $876.24.                10.  $14.875. 

12.  140  pwt.             21.  100  units. 

5.  $1036.351.            11.  $16.50. 

13.  3  sq.  yd.              22.  24  doz. 

6.  $291.84. 

14.  20  wk.                23.  72  doz. 

Page  215. 

15.  10  doz.                24.  800  cts. 
16.  144shts.             25.  75  cts. 

12.  $1.65.                  20.  40  wk. 

17.  3  cu.  yd.             26.  $1.00. 

13.  $19.12i.              21.  42  qt. 

14.  $4234.37£.          22.  $3.40. 

Page  233. 

15.  1J  Ib.                 23.  $148.43f  . 

16.  28yd.                 24.  44?\  mo. 

2.  1.  225  in.             11.  19,586  sec. 

17.  96  Ib.                  25.  $637.50. 

2.  500  in.             12.  77,009  sec. 

18.  $182.82.              26.  $13.485. 

3.  611  in.             13.  106',246sec. 

19.  18  wk. 

4.  1402  in.           14.  146,440  sec. 

5.  1783  in.           15.  8665  oz. 

Page  216. 

6.  63  pt.               16.  15,789  oz. 

27.  6£  cts.                 31.  $294f  . 

7.  117  pt.             17.  107,053  oz. 

28.  12doz.               32.  \  yr. 

8.  215  pt.             18.  173,546  oz. 

29.  $7.71f                33.  $360. 

9.  239  pt.             19.  336,170  oz. 

30.  1864  Ib.              34.  $1081.25. 

10.  467  pt.             20.  54,509  gr. 

3.  1.  37,858  gr.    11.  5923  in. 

Page  217. 
1.  $9.415. 

2.  68,915  gr.    12.  8883  sq.  in. 
3.  333  pt.          13.  13,965  sq.  in. 

Page  218. 

4.  1786  gi.        14.  18,156  sq.  yd. 

2.  $269.435.                4.  $2610.97. 

5.  491  pt.         15.  44,674  sq.  yd. 

3.  $1866.45. 

6.  6463  pt.       16.  89,328  sq.  yd. 

7.  1165m.        17.  159,072  cu.  in. 

Page  219. 

8.  1322m.       18.  246,201  cu.  in. 

6.  $65.57.                    8.  $21.86. 

9.  1889  in.       19.  483,850  cu.  in. 

7.  $48.89.                    9.  $164.51. 

10.  3613  in. 

Page  22O. 

Page  234. 

10.  $6917.55.            11.  $17.89. 

20.  12629.        26-  1996  sc. 

Page  231. 

21.  1675  9.        27.  10,776  min. 
22.  3143  sc.        28.  1997  in. 

3.  8pk.                     6.  24  gal. 

23.  2203  shts.    29.  33,636f  sq.rd. 

4.  320  sq.  rd.             7.  200  min. 

24.  2637  shts.     30.  548,658  in. 

5    180  in.                   8.  100  qr. 

25.  1111  pwt. 

ANSWERS 


277 


Page  235. 

1.  2  yd.  0  ft.  2£  in. 

2.  3  da.  21  hr.  20  min. 

3.  1  ft.  8f  in. 

4.  2  pk.  6.4  qt. 

5.  11  cwt.  20  Ib. 

6.  13  cu.  ft.  864  cu.  in. 

7.  53  rd.  1  yd.  2  ft.  6  in. 

8.  91  sq.  rd.  12  sq.  yd.  8  sq.  ft.  97f 

sq.  in. 

9.  5  da.  21  hr.  7  min.  12  sec. 

10.  1  qt.  1  pt.  1.5552  gi. 

11.  1  ft.  9.06  in. 

12.  12  K.  10  qr. 

13.  9  Gr.  4  doz. 

14.  63  53  19. 

15.  8  oz.  8  pwt. 

16.  17  cwt.  77  Ib.  14  oz. 

17.  2  pk.  5  qt.  0  pt.  2f  gi. 

18.  3  qt.  1  pt. 

19.  2  cd.  ft.  12.2368  cu.  ft. 

20.  19  sq.  rd.  22  sq.  yd.  4.554  sq.  ft. 

21.  293  rd.  1  yd.  2  ft.  6  in. 

22.  240  A. 

23.  11  cu.  ft.  432  cu.  in. 

24.  3  qt.  1  pt. 

25.  1  pk.  4  qt.  1.6  pt. 

26.  21|  gr. 

27.  5.76  gr. 

28.  25.55  da. 

29.  18.018  units. 

30.  16  qr.  16  sheets. 


Page  236. 

3.  1.  166  gal.  1  qt.  1  pt. 

2.  537  gal. 

3.  213  gal.  2  qt.  3  gi. 

4.  23  rd.  2  yd.  1  ft.  2  in. 

5.  18  rd.  3  in. 

6.  13  rd.  3  yd.  2  ft. 


7.  73  bu.  1  qt.  1  pt. 

8.  90  bu.  2  pk.  1  qt. 

9.  63  bu.  1  pk.  5  qt. 

10.  3  cwt.  92  Ib.  3  oz. 

11.  5  cwt.  77  Ib.  6  oz. 

12.  3  T.  94  Ib. 

13.  2  T.  8  cwt.  6  Ib. 

14.  32  wk.  20  hr. 

15.  2  hr.  34  min.  39  sec. 

16.  7  da.  13  hr.  45  min. 

.  1.  3  sq.  yd.  1  sq.  ft.  22  sq.  in. 

2.  4  sq.  yd.  6  sq.  ft. 

3.  6  R.  1  qr.  1  sheet. 

4.  7  R.  2  qr.  18  sheets. 

5.  5  cu.  ft.  335  cu.  in. 

6.  1  Ib.  4  oz.  7  pwt.  17  gr. 


7. 
8. 
9. 
10. 
6.  1. 
2. 
3. 
4. 
5. 
6. 
7. 


Page  237. 

1  Ib.  7  oz.  15  pwt.  17  gr. 
1  Ib.  93  23  29  9  gr. 
1C. 
9  A. 

*yL 


bu. 


f  wk. 


8. 

9. 
10. 
11. 

12.  m 

13.  4-H 


bbl. 


Page  238. 

2.  1.  63  gal.  1  qt. 

2.  61  bu.  1  pk.  1  qt. 

3.  26  da.  12  hr.  52  sec. 

4.  214  Ib.  3  oz. 

5.  68  Ib.  0  oz.  19  pwt.  3  gr. 

6.  88  yd.  1  ft.  2  in. 

7.  128  gal.  1  qt.  0  pt.  3  gi. 

8.  98  A.  96  sq.  rd. 


278 


ANSWERS 


Page  239. 

9.  42  T.  14  cwt.  68  Ib.  9  oz. 
10.  26  yd.  2  ft.  5  in. 

2.  4  bu.  3  pk.  7  qt. 
8.  4  bu.  0  pk.  6  qt. 
4.  6  gal.  3  qt.  1  pt.  3  gi. 
6.  2  da.  22  min.  20  sec. 

6.  32  Ib.  10  oz.  3  pwt. 

7.  20  yd.  1  ft.  9  in. 

8.  14  rd.  0  ft.  0  in. 

9.  15  Ib.  10  oz. 

10.  54  sq.  rd.  264  sq.  ft.  69  sq.  in. 


Page  24O. 

11.  14  T.  16  cwt  39  Ib.  14  oz. 

12.  3  gal.  1  qt.  1  pt.  2  gi. 
2.  1.  62  yr.  5  mo.  5  da. 

2.  15  yr.  6  mo.  8  da. 

3.  52  yr.  4  mo.  29  da. 

4.  10  yr.  10  mo.  17  da. 
6.  84  yr.  3  mo.  11  da. 


Page  241. 

2.  1.  45  bu.  2  pk.  0  qt. 

2.  47  gal.  0  qt.  0  pt.  1  gi. 

3.  59  Ib.  8  oz.  2  pwt.  9  gr. 

4.  70  Ib.  6  oz.  6  dr.  2  9  8  gr. 

5.  1  da.  3  hr.  4  min.  36  sec. 

6.  29  T.  19  cwt.  38  Ib.  14  oz. 

7.  38  rd.  5  yd.  2  ft.  2  in. 

8.  165  cu.  yd.  8  cu.  ft.  1670  cu.  in. 

9.  56  sq.  yd.  2  sq.  ft.  48  sq.  in. 

10.  31  da.  23  hr.  13  min. 

11.  216  rd.  3  yd.  2  ft.  3  in. 

12.  64  K.  17  qr.  2  sheets. 

13.  58  bbl.  30  gal. 

14.  32  yr.  3  mo.  18  da.  18  hr. 

15.  53  T.  3  cwt.  66  Ib.  4  oz.  8  dr. 


2.  1.  4  gal.  2  qt.  1  pt.  If  gi. 

2.  5  bu.  1  pk.  4  qt.  1  pt. 

3.  3  yd.  0  ft.  3|  in. 

4.  5  cwt.  79  Ib.  15£  oz. 

Page  242. 

6.  5  Ib.  4  oz.  llf  pwt. 

6.  31b.  8oz.  7  dr.  2f  J}. 

7.  2  hr.  53  min.  6f  sec. 

8.  6  sq.  yd.  5  sq.  ft.  40  sq.  in. 

9.  4  rd.  2  yd.  2  ft.  6£  in. 

10.  2  bbl.  1  gal.  1  qt.  If  pt. 

11.  20  bu.  2  pk.  5f  qt. 

12.  3  gal.  2  qt.  0  pt.  1-^  gi. 

13.  1  mi.  183  rd.  4  ft.  8T8T  in. 

14.  5  A.  46  sq.  rd.  21£  sq.  yd. 

15.  1  T.  15  cwt.  11  Ib.  8f  oz. 

Page  243. 

2.  432  sq.  ft. 

3.  300  sq.  ft. 

4.  204  sq.  ft. ;  22|  sq.  yd. 

5.  144  sq.  ft. ;  16  sq.  yd. 

6.  912  sq.  ft. ;  101£  sq.  yd. 

Page  244. 

7.  19,500  sq.  rd. ;  121|  A. 

8.  750  sq.  ft, ;  $10.42. 

9.  $32.30. 

10.  36  A.  16  sq.  rd. 

11.  $2025. 

12.  $48. 

13.  110.274  A. 

14.  $84. 

15.  $99,975. 

16.  49  sq.  rd. 

17.  ll^A. 

18.  $50.65. 

19.  10  A. 

20.  $36. 


ANSWERS 


279 


Page  246. 

4.  $53.40.                    8.  $1363.95. 

2.  1.  1120  cu.  ft.     6.  1200  cu.  ft. 

5.  $40.25.                    9.  56  quarts. 

2.  1080  cu.  in.     7.  1668£  cu.  ft. 
3.  540  cu.  yd.      8.   1431  cu.  ft. 

6.  $250;  $281.25.     10.  6  pupils. 
7.  10  cts.                   11.  2  words. 

4.  1650  cu.  ft.     9.  840|  cu.  ft. 
5.  665  cu.  ft.      10.  64,375  cu.  ft. 

Page  252. 

12.  $840.                   14.  $487.50. 

Page  247. 

13.  $296.10.               15.  $1915. 

3.  1.  $26.25.             4.  $864.00. 

Page  253. 

2.  $69.88.            5.  $5371.09. 

3«CC£    f)(* 

2.  1.  $15.                 11.  $1.68. 

.    4>OOO.ZO. 

2.  $15.                 12.  $4.42. 

4.  1.  $72.73.            4.  $5818.18. 

3.  $20.                 13.  $17.52. 

2.  $170.45.          5.  $15.91. 

4.  $18.                  14.  $13.08. 

3.  $135.07. 

5.  $70.                 15.  $12.49. 

5.  1.  $71.11.            4.  $360.19. 

6.  $44.                  16.  $29.17. 

2.  $33.33.            5.  $152.21. 

7.  $35.40.            17.  $27.35. 

3.  $152.00. 

8.  $401.52.           18.  $37.87. 

9.  $150.80.           19.  $874.28. 

Page  248. 

10.  $1366.08.         20.  $21.66. 

2-   1.  lOf  ft.                6.  90  ft. 

2.  11^  ft.                7.  333|  ft. 

Page  254. 

3.  10  ft.                  8.  216  ft. 

1.  $98.12.              6.  $90.00. 

4.  10  ft.                  9.  33£  ft. 

2.  $137.79.             7.  $227.36. 

5.  21  ft.                10.  6  ft. 

3.  $177.00.             8.  $141.29. 

3.  $3.24.                    6.  90£f  Ib. 

4.  $297.43.             9.  $176.08. 

4.  $72.                       6.  $15.00. 

6.  $308.41.           10.  $123.36. 

Page  249. 

Page  255. 

7.  402£ft. 

1.  $104.89.                 6.  $141.00. 

Page  251V 

2.  $164.20.                 7.  $79.352. 

3.  1.  $6.                    11.  $.39. 

3.  $57.50.                   8.  $77.98. 

2.  $14.                  12.  $.6265. 
3.  $24.                  13.  $.848. 

4.  $180.51.                  9.  $116.12. 
5.  $234.83.                10.  $68.67. 

4.  $40.                  14,  $.525. 
5.  $62.50.            15.  $3.0072. 

GENERAL  REVIEW. 

6.  77  sheep.         16.  56£  Ib. 

1.  1.  $776,475.2675. 

7.  1971  bu.          17.  1414.92  Ib. 

2.  $899,721.54. 

8.  1250  bbl.         18.  7260  apples. 

3.  $877,968.74. 

9.  18  men.           19.  4440  dolls. 

2.  828.023. 

10.  20  horses.        20.  632|  yr. 

3.  901.3006. 

280 


ANSWERS 


4.  52  da.  12  hr.  12  min. 

6.  106  bu.  1  pk. 

6.  73  Ib.  4  oz. 

7.  25  rd.  5  yd.  0|  ft. 

8.  224.82. 


10. 
11. 

12. 


13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 

22. 
23. 
24. 
25. 
126. 
27. 
28. 
29. 
30. 
81. 
32. 
33. 
34. 
35. 
36. 
37. 
38. 
39. 
40. 
41. 


653  cu.  yd.  8  cu.  ft.  1124  cu.  in. 
(1)  1979323;    (2)  948009;    (3) 

1439615. 
(1)%;   (2)  6ftj  (3)3ff;   (4) 

6ff;  (5)482.87;  (6)2120.35; 

(7)  38.953  ;  (8)  4240.7. 
484.006651. 
44  yr.  4  mo.  12  da. 
11  rd.  4  yd.  6  in. 
1  bu.  0  pk.  7  qt. 
5  gal.  0  qt.  1  pt. 
4.1722. 
17. 

1689.375. 
(1)  119173214;  (2)  229382875; 

(3)  149155212. 


6.81096;  73.08225;  .03396. 

$14.50. 

$1944.39. 

167  mi.  165  rd.  6£  ft. 

$46.67. 


92  hhd.  57  gal.  1  qt.  1  pt. 


21;  12360;  43647.9. 

3  mi.  232  rd.  11£  ft.  ;  8  cd.  30f  ft. 


80bbl. 

35  da. 

200  C. 

£. 

A;  120;  28. 


42. 

43. 

44. 


46. 

47. 

48. 
49. 

50. 


51. 
52. 
63. 
64. 
65. 

56. 
57. 
58. 
69. 
60. 
61. 
62. 
63. 
64. 
65. 
66. 
67. 
68. 
69. 
70. 
71. 
72. 
73. 
74. 
75. 
76. 
77. 
78. 


75. 

TV;  m;  Hfi;  f- 

5,  59;  2,  2,  139;  23,  3*;  none; 
5,  13,  23. 

if;  m»  T^V 

33f;  82&;  62^  ;  ISff  ;  444. 


4196  gi.  ;  2580  pt. 

13  mi.  109  rd.  2  yd.  1  ft.  10  in.  ; 

37037  cu.  yd.  1  cu.  ft. 
17  cwt.  77  Ib.  12f  oz.  ;  3  pk.  ; 

518  A.  64  sq.  rd.  ;  9  hr.  14 

min.  24  sec. 
42|. 

76.5466. 
.3706883. 

87.45;  874.5;  87450;  874600. 
.008745;  .0008745;  .000008745; 

.0000008745. 

.04965  -f;  1000000000;  .08334-. 
13  bbl. 
$52.437. 
4hr. 
$2.25. 
39f  A. 
751£  cu.  ft. 
15|f  cd. 

7,762,392  cu.  in. 
1013£  bd.  ft. 
2.453  A. 
988sqvft. 
$3246.99. 
63  bd.  ft. 
$22.28. 
$209.44. 
$.86. 
$1.73. 
$963.77. 
$515.20. 
$159.22. 

$65.17;  $566.17. 
$5.00;  $5.50. 


ANSWERS 


281 


79.  $105.41. 

80.  $120.38. 

81.  1463  mi. 

82.  A.  55,  B.  54,  C.  100. 

83.  307. 

84.  27,735. 

85.  55,111. 

86.  1584. 

87.  850  bu. 

88.  $20,000;  $12,500. 

89.  5T5Tda, 

90.  157  A.  144  sq.  rd. 

91.  $443.30. 


92.  $52.54. 

93.  45. 

94.  64  rd. 

95.  249;  332. 

96.  $45.00. 

97.  $31627. 

98.  7/T. 

99.  $5.62f 

100.  B.  =-',  B.  = 


Int. 


;  R   = 


Time 


Int. 


Int. 
B.  x  B/ 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 

AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  SO  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $1.OO  ON  THE  SEVENTH  DAY 
OVERDUE. 


SEP  131935 


JAN- 


LD  21-100m-7,'33 


YB   17421 


8005G3 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


